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References
The bibliography uses the following abbreviations:
arXiv = The arXiv eprint archive at http://arXiv.org/DCC = Designs, Codes and CryptographyDM = Discrete MathematicsJCT = Journal of Combinatorial TheoryPGIT = IEEE Transactions on Information Theory
1. M. Aaltonen, Linear programming bounds for tree codes, PGIT 25 (1979),85–90.
2. M. Aaltonen, A new upper bound on nonbinary block codes, DM 83 (1990),139–160.
3. A. V. Alekseevskii, Finite commutative Jordan subgroups of complex simpleLie groups, Functional Anal. Appl. 8 (1974), 277–279.
4. O. Amrani and Y. Beery, Reed-Muller codes: projections onto GF(4) andmultilevel construction, PGIT 47 (2001), 2560–2565.
5. O. Amrani, Y. Beery and A. Vardy, Bounded-distance decoding of the Leechlattice and the Golay code, in Algebraic Coding (Paris, 1993), Lecture NotesComput. Sci. 781 (1994), 236–248.
6. O. Amrani, Y. Beery, A. Vardy, F.-W. Sun and H. C. A. van Tilborg, TheLeech lattice and the Golay code: bounded-distance decoding and multilevelconstructions, PGIT 40 (1994), 1030–1043.
7. J. B. Anderson, Digital Transmission Engineering , IEEE Press and Prentice-Hall, NY, 1998.
8. J. L. Anderson, On minimal decoding sets for the extended binary Golay code,PGIT 38 (1992), 1560–1561.
9. A. N. Andrianov, Quadratic Forms and Hecke Operators, Springer, 1987.10. T. Aoki, P. Gaborit, M. Harada, M. Ozeki and P. Sole, On the covering radius
of Z4-codes and their lattices, PGIT 45 (1999), 2162–2168.11. K. T. Arasu and T. A. Gulliver, Self-dual codes over Fp and weighing matrices,
PGIT 47 (2001) 2051–2055.
392 References
12. M. Araya and M. Harada, MDS codes over F9 related to the ternary Golaycode, DM 282 (2004) 233–237.
13. A. Ashikhmin and E. Knill, Nonbinary quantum stabilizer codes, PGIT 47(2001) 3065–3072.
14. A. Ashikhmin and S. Litsyn, Upper Bounds on the size of quantum codes,PGIT 45 (1999), 1206–1216.
15. E. F. Assmus, Jr., H. F. Mattson, Jr. and R. J. Turyn, Research to developthe algebraic theory of codes, Report AFCRL-67-0365, Air Force CambridgeRes. Labs., Bedford, MA, June 1967.
16. E. F. Assmus, Jr. and V. S. Pless, On the covering radius of extremal self-dualcodes, PGIT 29 (1983), 359–363.
17. A. O. L. Atkin and J. Lehner, Hecke operators on Γ0(m), Math. Ann. 185(1970), 134–160.
18. A. Baartmans and V. Y. Yorgov, Some new extremal codes of lengths 76 and78, PGIT 49 (2003), 1353–1354.
19. C. Bachoc, Applications of coding theory to the construction of modular lat-tices, JCT A 78 (1997), 92–119.
20. C. Bachoc, On harmonic weight enumerators of binary codes, DCC 18 (1999),11–28.
21. C. Bachoc, Harmonic weight enumerators of non-binary codes andMacWilliams identities, in Codes and association schemes (Piscataway, NJ,1999), DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 56, Amer. Math.Soc., Providence, RI, 2001; pp. 1–23.
22. C. Bachoc, Designs, groups and lattices, J. Theorie Nombres Bordeaux, toappear, 2005.
23. C. Bachoc and P. Gaborit, On extremal additive F4 codes of length 10 to 18, inInternational Workshop on Coding and Cryptography (Paris, 2001), Electron.Notes Discrete Math. 6 (2001), 10 pp.
24. C. Bachoc and P. Gaborit, Designs and self-dual codes with long shadows,JCT A 105 (2004), 15–34.
25. B. Bajnok, Construction of spherical t-designs, Geom. Dedicata, 43 (1992),167–179.
26. B. Bajnok, Construction of spherical 3-designs, Graphs and Combinatorics, 14(1998), 97–107.
27. A. Bak, K-Theory of Forms, Princeton Univ. Press, 1981.28. E. Bannai, S. T. Dougherty, M. Harada and M. Oura, Type II codes, even
unimodular lattices and invariant rings, PGIT 45 (1999), 1194–1205.29. E. S. Barnes and G. E. Wall, Some extreme forms defined in terms of Abelian
groups, J. Australian Math. Soc. 1 (1959), 47–63.30. H.-J. Bartels, Zur Galoiskohomologie definiter arithmetischer Gruppen, J.
Reine Angew. Math. 298 (1978), 89–97.31. G. F. M. Beenker, A note on extended quadratic residue codes over GF (9) and
their ternary images, PGIT 30 (1984), 403–405.32. C. H. Bennett, D. DiVincenzo, J. A. Smolin and W. K. Wootters, Mixed
state entanglement and quantum error correction, Phys. Rev. A 54 (1996),3824–3851.
33. D. J. Benson, Polynomial Invariants of Finite Groups, Cambridge Univ. Press,1993.
34. E. R. Berlekamp, F. J. MacWilliams and N. J. A. Sloane, Gleason’s theoremon self-dual codes, PGIT 18, (1972), 409–414.
References 393
35. K. Betsumiya, On the classification of Type II codes over F2r with binarylength 32, Preprint, 2002.
36. K. Betsumiya and Y.-J. Choie, Jacobi forms over totally real fields and type IIcodes over Galois rings GR(2m, f), European J. Combin. 25 (2004), 475–486.
37. K. Betsumiya and Y.-J. Choie, Codes over F4, Jacobi forms and Hilbert-Siegelmodular forms over Q(
√5), European J. Combin. 26 (2005), 629–650.
38. K. Betsumiya, S. Georgiou, T. A. Gulliver, M. Harada and C. Koukouvinos,On self-dual codes over some prime fields, DM 262 (2003), 37–58.
39. K. Betsumiya, T. A. Gulliver and M. Harada, Binary optimal linear rate 1/2codes, in Applied Algebra, Algebraic Algorithms and Error-Correcting Codes(Honolulu, HI, 1999), Lecture Notes Comput. Sci. 1719 (1999), pp. 462–471.
40. K. Betsumiya, T. A. Gulliver and M. Harada, On binary extremal formallyself-dual even codes, Kyushu J. Math. 53 (1999), 421–430.
41. K. Betsumiya, T. A. Gulliver and M. Harada, Extremal self-dual codes overF2 × F2, DCC 28 (2003), 171–186.
42. K. Betsumiya, T. A. Gulliver, M. Harada and A. Munemasa, On type II codesover F4, PGIT 47 (2001), 2242–2248.
43. K. Betsumiya and M. Harada, Binary optimal odd formally self-dual codes,DCC 23 (2001), 11–21.
44. K. Betsumiya and M. Harada, Classification of formally self-dual even codesof lengths up to 16, DCC 23 (2001), 325–332.
45. K. Betsumiya and M. Harada, Optimal self-dual codes over F2 × F2 withrespect to the Hamming weight, PGIT 50 (2004), 356–358.
46. K. Betsumiya, M. Harada and A. Munemasa, Type II codes over F2r , inApplied Algebra, Algebraic Algorithms and Error-Correcting Codes (Melbourne,2001), Lecture Notes Comput. Sci. 2227 (2001), pp. 102-111.
47. K. Betsumiya, S. Ling and F. R. Nemenzo, Type II codes over F2m + uF2m ,DM 275 (2004), 43–65.
48. V. K. Bhargava and C. Nguyen, Circulant codes based on the prime 29, PGIT26 (1980), 363–364.
49. R. T. Bilous, Enumeration of binary self-dual codes of length 34, J. Combin.Math. Combin. Computing, to appear, 2005.
50. R. T. Bilous and G. H. J. van Rees, An enumeration of binary self-dual codesof length 32, DCC 26 (2002), 61–86.
51. I. F. Blake, Properties of generalized Pless codes, in Proc. 12th Allerton Conf.Circuit and System Theory, Univ. Ill., Urbana, 1974, pp. 787–789.
52. I. F. Blake, On a generalization of the Pless symmetry codes, Inform Control27 (1975), 369–373.
53. M. Blaum and J. Bruck, Decoding the Golay code with Venn diagrams, PGIT36 (1990), 906–910.
54. F. van der Blij, An invariant of quadratic forms mod 8, Indag. Math. 21 (1959),291–293.
55. S. Bocherer, On the notion of extremal modular forms and analytically ex-tremal lattices, Preprint, 2005.
56. S. Bocherer, Siegel modular forms and theta series, in Theta functions (Bow-doin 1987), Proc. Sympos. Pure Math., 49, Part 2, Amer. Math. Soc., Provi-dence, RI, 1989, pp. 3–17.
57. B. Bolt, The Clifford collineation, transform and similarity groups III: gener-ators and involutions, J. Australian Math. Soc. 2 (1961), 334–344.
394 References
58. B. Bolt, T. G. Room and G. E. Wall, On Clifford collineation, transform andsimilarity groups I, J. Australian Math. Soc. 2 (1961), 60–79.
59. B. Bolt, T. G. Room and G. E. Wall, On Clifford collineation, transform andsimilarity groups II, J. Australian Math. Soc. 2 (1961), 80–96.
60. A. Bonnecaze, A. R. Calderbank and P. Sole, Quaternary quadratic residuecodes and unimodular lattices, PGIT 41 (1995), 366–377.
61. A. Bonnecaze, Y.-J. Choie, S. T. Dougherty and P. Sole, Splitting the shadow,DM 270 (2003), 43–60.
62. A. Bonnecaze, A. Desidiri Bracco, S. T. Dougherty, L. R. Nochefranca and P.Sole, Cubic self-dual binary codes, PGIT 49 (2003), 2253–2259.
63. A. Bonnecaze, P. Gaborit, M. Harada, M. Kitazume and P. Sole, Niemeierlattices and Type II codes over Z4, DM 205 (1999), 1–21.
64. A. Bonnecaze, B. Mourrain and P. Sol’e, Jacobi polynomials, type II codes,and designs, DCC 16 (1999), 215–234.
65. A. Bonnecaze, E. M. Rains and P. Sole, 3-colored 5-designs and Z4-codes, J.Statist. Plann. Inference 86 (2000), 349–368.
66. A. Bonnecaze, P. Sole, C. Bachoc and B. Mourrain, Type II codes over Z4,PGIT 43 (1997), 969–976.
67. R. E. Borcherds, Automorphic forms with singularities on Grassmannians, In-vent. Math. 132 (1998), 491–562 [arXiv: alg-geom/9609022].
68. W. Bosma, J. Cannon and C. Playoust, The Magma algebra system I: Theuser language, J. Symb. Comp., 24 (1997), 235–265.
69. M. Bossert, On decoding binary quadratic residue codes, in Applied algebra,algebraic algorithms and error-correcting codes (Menorca, 1987), Lecture NotesComput. Sci. 356 (1989), 60–68.
70. I. Bouyukliev, S. Bouyuklieva, T. A. Gulliver and P. R. J. Ostergard, Clas-sification of optimal binary self-orthogonal codes, J. Combin. Math. Combin.Computing, to appear, 2005.
71. I. Bouyukliev and P. R. J. Ostergard, Classification of self-orthogonal codesover F3 and F4, SIAM J. Discrete Mathematics, to appear, 2005.
72. S. Bouyuklieva, New extremal self-dual codes of lengths 42 and 44, PGIT 43(1997), 1607–1612.
73. S. Bouyuklieva, On the binary self-dual codes with an automorphism of order2, DCC 12 (1997), 39–48.
74. S. Bouyuklieva, On the automorphisms of order 2 with fixed points for theextremal self-dual codes of length 24m, DCC 25 (2002), 5–13.
75. S. Bouyuklieva, On the automorphism group of a doubly-even (72, 36, 16)code, PGIT 50 (2004), 544–547.
76. S. Bouyuklieva, Some optimal self-orthogonal and self-dual codes, DM 287(2004), 1–10.
77. S. Bouyuklieva and I. Bouyukliev, Extremal self-dual codes with an automor-phism of order 2, PGIT 44 (1998), 323–328.
78. S. Bouyuklieva and M. Harada, On type IV self-dual codes over Z4, DM 247(2002), 25–50.
79. S. Bouyuklieva and M. Harada, Extremal self-dual [50, 25, 10] codes withautomorphisms of order 3 and quasi-symmetric 2-(49, 9, 6) designs, DCC 28(2003), 163–169.
80. S. Bouyuklieva and V. Y. Yorgov, Singly-even self-dual codes of length 40,DCC 9 (1996), 131–141.
References 395
81. H. Braun, Geschlecter quadratischer Formen, J. Reine Angew. Math. 182(1940), 32–49.
82. M. Broue and M. Enguehard, Polynomes des poids de certains codes et fonc-tions theta de certains reseaux, Ann. scient. Ec. Norm. Sup. 4e serie, 5 (1972),157–181.
83. M. Broue and M. Enguehard, Une famille infinie de formes quadratiquesentieres; leurs groupes d’automorphismes, Ann. scient. Ec. Norm. Sup. 4e serie,6 (1973), 17–52. Summary in C. R. Acad. Sc. Paris 274 (1972), 19–22.
84. A. E. Brouwer, Bounds on the size of linear codes, Chapter 4 of [426], pp.295–461.
85. A. E. Brouwer, A. M. Cohen and A. Neumaier, Distance-Regular Graphs,Springer, 1989.
86. R. A. Brualdi and V. S. Pless, Weight enumerators of self-dual codes, PGIT37 (1991), 1222–1225.
87. J. H. Bruinier, Borcherds products on O(2, l) and Chern classes of Heegnerdivisors, Lecture Notes Math. 1780, 2002.
88. F. C. Bussemaker and V. D. Tonchev, New extremal doubly-even codes oflength 56 derived from Hadamard matrices of order 28, DM 76 (1989), 45–49.
89. F. C. Bussemaker and V. D. Tonchev, Extremal doubly-even codes of length40 derived from Hadamard matrices, DM 82 (1990), 317–321.
90. A. R. Calderbank, P. J. Cameron, W. M. Kantor and J. J. Seidel, Z4-Kerdock codes, orthogonal spreads and extremal Euclidean line-sets, Proc.London Math. Soc. 75 (1997), 436–480.
91. A. R. Calderbank, A. R. Hammons, Jr., P. V. Kumar, N. J. A. Sloane and P.Sole, A linear construction for certain Kerdock and Preparata codes, Bull.Amer. Math. Soc. 29 (1993), 218–222.
92. A. R. Calderbank, R. H. Hardin, E. M. Rains, P. W. Shor and N. J. A. Sloane,A group-theoretic framework for the construction of packings in Grassmannianspaces, J. Algebraic Combin. 9 (1999), 129–140 [arXiv: math.CO/0208002].
93. A. R. Calderbank and W. M. Kantor, The geometry of two-weight codes, Bull.London Math. Soc., 118 (1986), 97–122.
94. A. R. Calderbank, W.-C. W. Li and B. Poonen, A 2-adic approach to theanalysis of cyclic codes, PGIT 43 (1997), 977–986.
95. A. R. Calderbank, E. M. Rains, P. W. Shor and N. J. A. Sloane, Quantumerror correction and orthogonal geometry, Phys. Rev. Lett. 78 (1997), 405–409[arXiv: quant-ph/9605005].
96. A. R. Calderbank, E. M. Rains, P. W. Shor and N. J. A. Sloane, Quantumerror correction via codes over GF(4), PGIT 44, (1998), 1369–1387 [arXiv:quant-ph/9608006].
97. A. R. Calderbank and P. W. Shor, Good quantum error-correcting codes exist,Phys. Rev. A, 54, pp. 1098–1105 (1996) [arXiv: quant-ph/9512032].
98. A. R. Calderbank and N. J. A. Sloane, Modular and p-adic cyclic codes, DCC6 (1995), 21–35 [arXiv: math.CO/0311319].
99. A. R. Calderbank and N. J. A. Sloane, Double circulant codes over Z4 andeven unimodular lattices, J. Algebraic Combinatorics 6 (1997), 119–131.
100. J. Cannon et al., The Magma Computational Algebra System for Algebra, Num-ber Theory and Geometry, published electronically at http://magma.maths.usyd.edu.au/magma/.
101. J.-C. Carlach and A. Otmani, A systematic construction of self-dual codes,PGIT 49 (2003), 3005–3009.
396 References
102. C. Carlet, Z2k -linear codes, PGIT 44 (1998), 1543–1547.103. J. W. S. Cassels, Rational Quadratic Forms, Academic Press, NY, 1978.104. R. Chapman, S. T. Dougherty, P. Gaborit and P. Sole, 2−modular lattices
from ternary codes, J. Theorie Nombres Bordeaux 14 (2002), 1–13.105. G. Chen and R. K. Brylinski, eds., Mathematics of Quantum Computation,
Chapman and Hall, NY, 2002.106. Y. Cheng and R. Scharlau, personal communication, Sept., 1987.107. Y. Cheng and N. J. A. Sloane, The automorphism group of an [18,9,8] quater-
nary code, DM 83 (1990), 205–212.108. C. Chevalley, Invariants of finite groups generated by reflections, Amer. J.
Math. 67 (1955), 778–782.109. Y.-J. Choie and S. T. Dougherty, Codes over Σ2m and Jacobi forms over the
quaternions, Appl. Algebra Engrg. Comm. Comput. 15 (2004), 129–147.110. Y.-J. Choie and S. T. Dougherty, Codes over rings, complex lattices and
Hermitian modular forms, European J. Combin. 26 (2005), 145–165.111. Y.-J. Choie, S. T. Dougherty and H. Kim, Complete joint weight enumerators
and self-dual codes, PGIT 49 (2003), 1275-1282.112. Y.-J. Choie and E. Jeong, Jacobi forms over totally real fields and codes over
Fp, Illinois J. Math. 46 (2002), 627–643.113. Y.-J. Choie and N. Kim, The complete weight enumerator of type II codes
over Z2m and Jacobi forms, PGIT 47 (2001), 396–399.114. Y-J. Choie, B. Mourrain and P. Sole, Rankin-Cohen brackets and invariant
theory, J. Algebraic Combinatorics 13 (2001), 5–13.115. Y-J. Choie and P. Sole, A Gleason formula for Ozeki polynomials, JCT A 98
(2002), 60–73.116. Y-J. Choie and P. Sole, Self-dual codes over Z4 and half-integral weight mod-
ular forms, Proc. Amer. Math. Soc. 130 (2002), 3125–3131.117. Y-J. Choie and P. Sole, Ternary codes and Jacobi forms, DM 282 (2004),
81–87.118. R. Cleve, Quantum stabilizer codes and classical linear codes, Phys. Rev. A 55
(1997), 4054–4059.119. H. Cohen, A Course in Computational Algebraic Number Theory, Springer,
1996.120. J. H. Conway, R. H. Hardin and N. J. A. Sloane, Packing lines, planes, etc.:
packings in Grassmannian space, Experimental Math. 5 (1996), 139–159 [arXiv:math.CO/0208004].
121. J. H. Conway and V. S. Pless, On the enumeration of self-dual codes, JCT A28 (1980), 26–53.
122. J. H. Conway and V. S. Pless, On primes dividing the group order of a doubly-even (72, 36, 16) code and the group of a quaternary (24, 12, 10) code, DM 38(1982), 157–162.
123. J. H. Conway, V. S. Pless and N. J. A. Sloane, Self-dual codes over GF (3)and GF (4) of length not exceeding 16, PGIT 25 (1979), 312–322.
124. J. H. Conway, V. S. Pless and N. J. A. Sloane, The binary self-dual codes oflength up to 32: A revised enumeration, JCT A 60 (1992), 183–195.
125. J. H. Conway and N. J. A. Sloane, On the enumeration of lattices of determi-nant one, J. Number Theory 15 (1982), 83–94.
126. J. H. Conway and N. J. A. Sloane, Soft decoding techniques for codes andlattices, including the Golay code and the Leech lattice, PGIT 32 (1986), 41–50.
References 397
127. J. H. Conway and N. J. A. Sloane, Low-dimensional lattices I: Quadratic formsof small determinant, Proc. Royal Soc. A 418 (1988), 17–41.
128. J. H. Conway and N. J. A. Sloane, Low-dimensional lattices II: Subgroups ofGL(n, Z), Proc. Royal Soc. A 419 (1988), 29–68.
129. J. H. Conway and N. J. A. Sloane, A new upper bound for the minimum of anintegral lattice of determinant one, Bull. Amer. Math. Soc. 23 (1990), 383–387;erratum: 24 (1991), 479.
130. J. H. Conway and N. J. A. Sloane, A new upper bound on the minimal distanceof self-dual codes, PGIT 36 (1990), 1319–1333.
131. J. H. Conway and N. J. A. Sloane, Self-dual codes over the integers modulo 4,JCT A 62 (1993), 30–45.
132. J. H. Conway and N. J. A. Sloane, On lattices equivalent to their duals, J.Number Theory 48 (1994), 373–382.
133. J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups,Springer, 1998, 3rd. ed., 1998.
134. D. Coppersmith and G. Seroussi, On the minimum distance of some quadraticresidue codes, PGIT 30 (1984), 407–411.
135. H. S. M. Coxeter, Regular Complex Polytopes Cambridge Univ. Press, 2nd. ed.,1991.
136. D. B. Dalan, New extremal type I codes of lengths 40, 42 and 44, DCC 30(2003), 151–157.
137. D. B. Dalan, New extremal binary [44, 22, 8] codes, PGIT 49 (2003), 747–748.138. D. B. Dalan, Extremal binary self-dual codes of lengths 42 and 44 with new
weight enumerators, Kyushu J. Math. 57 (2003), 333–345.139. L. E. Danielsen and M. G. Parker, On the classification of all self-dual additive
codes over GF(4) of length up to 12, Preprint, 2005.140. F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied
Tables, Cambridge Univ. Press, 1966.141. E. Dawson, Self-dual ternary codes and Hadamard matrices, Ars Comb. 19A
(1985), 303–308.142. P. Delsarte, Bounds for unrestricted codes, by linear programming, Philips Res.
Reports 27 (1972), 272–289.143. P. Delsarte, J.-M. Goethals and J. J. Seidel, Spherical codes and designs,
Geom. Dedicata, 6 (1977), 363–388.144. H. Derksen and G. Kemper, Computational Invariant Theory, Springer, 2002.145. S. M. Dodunekov, V. A. Zinoviev and J. E. M. Nilsson, On the algebraic
decoding of some maximal quaternary codes and binary Golay codes (Russian),Probl. Pered. Inform. 35 (1999), no. 4, 59–67; Problems Inform. Transmission35 (1999), no. 4, 338–345.
146. R. Dontcheva, New binary [70, 35, 12] self-dual and binary [72, 36, 12] self-dualdoubly-even codes, Serdica Math. J. 27 (2001), 287–302.
147. R. Dontcheva and M. Harada, New extremal self-dual codes of length 62 andrelated extremal self-dual codes, PGIT 48 (2002), 2060–2064.
148. R. Dontcheva and M. Harada, Some extremal self-dual codes with an auto-morphism of order 7, Appl. Algebra Engrg. Comm. Comput. 14 (2003), 75–79.
149. R. Dontcheva and M. Harada, Extremal doubly-even [80,40,16] codes with anautomorphism of order 19, Finite Fields Appl. 9 (2003), 157–167.
150. R. Dontcheva, A. van Zanten and S. Dodunekov, Binary self-dual codes withautomorphisms of composite order, PGIT 50 (2004), 311–318.
398 References
151. S. T. Dougherty, Shadow codes and weight enumerators, PGIT 41 (1995),762–768.
152. S. T. Dougherty, P. Gaborit, M. Harada, A. Munemasa and P. Sole, Type IVself-dual codes over rings, PGIT 45 (1999), 2345–2360.
153. S. T. Dougherty, T. A. Gulliver and M. Harada, Extremal binary self-dualcodes, PGIT 43 (1997), 2036–2046.
154. S. T. Dougherty, T. A. Gulliver and M. Harada, Type II self-dual codes overfinite rings and even unimodular lattices, J. Alg. Combin. 9 (1999), 233–250.
155. S. T. Dougherty and M. Harada, Shadow optimal self-dual codes, Kyushu J.Math. 53 (1999), 223–237.
156. S. T. Dougherty and M. Harada, New extremal self-dual codes of length 68,PGIT 45 (1999), 2133–2136.
157. S. T. Dougherty, M. Harada, P. Gaborit and P. Sole, Type II codes overF2 + uF2, PGIT 45 (1999), 32–45.
158. S. T. Dougherty, M. Harada and M. Oura, Note on the g-fold joint weightenumerators of self-dual codes over Zk, Appl. Algebra Engrg. Comm. Comput.11 (2001), 437–445.
159. S. T. Dougherty, M. Harada and P. Sole, Shadow lattices and shadow codes,DM 219 (2000), 49–64.
160. S. T. Dougherty, M. Harada and P. Sole, Shadow codes over Z4, Finite FieldsApplic. 7 (2001), 507–529.
161. W. Duke, On codes and Siegel modular forms, Internat. Math. Res. Notices 5(1993), 125–136.
162. I. M. Duursma, M. Greferath and S. E. Schmidt, On the optimal Z4 codes oftype II and length 16, JCT A 92 (2000), 77–82.
163. H. Dym and H. P. McKean, Fourier Series and Integrals, Academic Press,NY, 1972.
164. W. Ebeling Lattices and Codes, Vieweg, Braunschweig/Wiesbaden, 2nd ed.,2002.
165. W. Eholzer and N. P. Skoruppa, Modular invariance and uniqueness of con-formal characters, Commun. Math. Phys. 174 (1995) 117–136.
166. M. Eichler and D. Zagier, The Theory of Jacobi Forms, Birkhauser, Boston,1985.
167. N. D. Elkies, Lattices and codes with long shadows, Math. Res. Lett., 2 (1995),643–651.
168. N. D. Elkies, Lattices, linear codes, and invariants, Notices Amer. Math. Soc.47 (2000), 1238–1245 and 1382–1391.
169. M. Esmaeili, T. A. Gulliver and A. K. Khandani, On the Pless-constructionand ML decoding of the (48, 24, 12) quadratic residue code, PGIT 49 (2003),1527–1535.
170. C. Faith, Algebra II: Ring Theory, Springer, 1976.171. F. Fekri, S. W. McLaughlin, R. M. Mersereau and R. W. Schafer, Double
circulant self-dual codes using finite-field wavelet transforms, in Applied Alge-bra, Algebraic Algorithms and Error-Correcting Codes (Honolulu, HI, 1999),Lecture Notes Comput. Sci. 1719 (1999), pp. 355–364.
172. X.-N. Feng and Z.-D. Dai, Notes on finite geometries and the construction ofPBIB designs, V: Some “Anzahl” theorems in orthogonal geometry over finitefields of characteristic 2, Sci. Sinica 13 (1964), 2005-2008.
173. J. E. Fields, Tables of Indecomposable Z/(4) Codes, published electronically athttp://www.math.uic.edu/∼fields/z4/.
References 399
174. J. E. Fields, P. Gaborit, J. S. Leon and V. S. Pless, All self-dual Z4 codes oflength 15 or less are known, PGIT 44 (1998), 311–322.
175. G. D. Forney, Jr., N. J. A. Sloane and M. D. Trott, The Nordstrom-Robinsoncode is the binary image of the octacode, in Coding and Quantization: DI-MACS/IEEE Workshop October 19–21, 1992, ed. R. Calderbank, G. D. For-ney, Jr. and and N. Moayeri, Amer. Math. Soc. (1993), pp. 19–26.
176. E. Freitag, Siegelsche Modulfunktionen, Springer, 1983.177. A. Frohlich and M. J. Taylor, Algebraic Number Theory, Cambridge Univ.
Press, 1991.178. P. Gaborit, Mass formulas for self-dual codes over Z4 and Fq +uFq rings, PGIT
42 (1996), 1222–1228.179. P. Gaborit, Quadratic double circulant codes over fields, JCT A 97 (2002)
85–107.180. P. Gaborit, Tables of Self-Dual Codes, published electronically at
http://www.unilim.fr/pages perso/philippe.gaborit/SD/, 2004.181. P. Gaborit, A bound for certain s-extremal lattices and codes, Preprint, 2005.182. P. Gaborit and M. Harada, Construction of extremal Type II codes over Z4,
DCC, submitted.183. P. Gaborit, W. C. Huffman, J.-L. Kim and V. S. Pless, On additive GF(4)
codes, in Codes and Association Schemes (Piscataway, NJ, 1999), ed. A. Bargand S. Litsyn, Amer. Math. Soc., Providence, RI, 2001, pp. 135–149.
184. P. Gaborit, J.-L. Kim and V. S. Pless, Decoding binary R(2, 5) by hand, DM264 (2003), 55–73.
185. P. Gaborit and A. Otmani, Experimental construction of self-dual codes, FiniteFields Appl. 9 (2003), 372–394.
186. P. Gaborit, V. S. Pless, P. Sole and A. O. L. Atkin, Type II codes over F4,Finite Fields Applic. 8 (2002), 171–183.
187. M. Gaulter, Minima of odd unimodular lattices in dimension 24m, J. NumberTheory 91 (2001), 81–91.
188. S. Georgiou and C. Koukouvinos, Self-dual codes over GF (7) and orthogonaldesigns, Utilitas Math. 60 (2001) 79–89.
189. S. Georgiou and C. Koukouvinos, MDS self-dual codes over large prime fields,Finite Fields Appl. 8 (2002), 455–470.
190. S. P. Glasby, On the faithful representations, of degree 2n, of certain extensionsof 2-groups by orthogonal and symplectic groups. J. Australian Math. Soc. Ser.A 58, (1995), 232–247.
191. A. M. Gleason, Weight polynomials of self-dual codes and the MacWilliamsidentities, in Actes, Congres International de Mathematiques (Nice, 1970),Gauthiers-Villars, Paris, 1971, Vol. 3, pp. 211–215.
192. J.-M. Goethals and J. J. Seidel, Spherical designs, in Relations Between Com-binatorics and Other Parts of Mathematics, ed. D. K. Ray-Chaudhuri, Proc.Symp. Pure Math. 34 (1979), 255–272.
193. J.-M. Goethals and J. J. Seidel, Cubature formulae, polytopes and sphericaldesigns, in The Geometric Vein: The Coxeter Festschrift, ed. C. Davis et al.,Springer, 1981, pp. 203–218.
194. J.-M. Goethals and J. J. Seidel, The football, Nieuw Archief voor Wiskunde29 (1981), 50–58. Reprinted in Geometry and Combinatorics: Selected Worksof J. J. Seidel, ed. D. G. Corneil and R. Mathon, Academic Press, NY, 1991,pp. 363–371.
400 References
195. I. J. Good, Generalizations to several variables of Lagrange’s expansion, withapplications to stochastic processes, Proc. Camb. Phil. Soc. 56 (1960), 367–380.
196. D. M. Gordon, Minimal permutation sets for decoding the binary Golay codes,PGIT 28 (1982), 541–543.
197. D. Gottesman, A class of quantum error-correcting codes saturating the quan-tum Hamming bound, Phys. Rev. A 54 (1996), 1862–1868 [arXiv: quant-ph/9604038].
198. M. Grassl, Bounds on dmin for additive [[n, k, d]] QECC, published electroni-cally at http://iaks-www.ira.uka.de/home/grassl/QECC/index.html.
199. M. Greferath and S. E. Schmidt, Linear codes and rings of matrices, in Ap-plied Algebra, Algebraic Algorithms and Error-Correcting Codes (Honolulu, HI,1999), Lecture Notes Comput. Sci. 1719 (1999), pp. 160–169.
200. M. Greferath and S. E. Schmidt, Gray isometries for finite chain rings and anonlinear ternary (36, 312, 15) code, PGIT 45 (1999), 2522–2524.
201. M. Greferath and S. E. Schmidt, Finite-ring combinatorics and MacWilliams’equivalence theorem, JCT A 92 (2000), 17–28.
202. M. Greferath and U. Vellbinger, Efficient decoding of Zpk -linear codes, PGIT44 (1998), 1288–1291.
203. M. Greferath and E. Viterbo, On Z4- and Z9-linear lifts of the Golay codes,PGIT 45 (1999), 2524–2527.
204. R. L. Griess, Automorphisms of extra special groups and nonvanishing degree2 cohomology, Pacific J. Math. 48 (1973), 403–422.
205. P. Griffiths and J. Harris, Principles of Algebric Geometry Wiley, NY, 1978.206. B. Gross and G. Nebe, Globally maximal arithmetic groups, J. Algebra 272
(2004), 625–642.207. T. A. Gulliver, Optimal double circulant self-dual codes over F4, PGIT 46
(2000), 271–274.208. T. A. Gulliver and V. K. Bhargava, Self-dual codes based on the twin prime
product 35, Appl. Math. Lett. 5 (1992), 95–98.209. T. A. Gulliver and M. Harada, Weight enumerators of extremal singly-even
[60, 30, 12] codes, PGIT 42 (1996), 658–659.210. T. A. Gulliver and M. Harada, Classification of extremal double circulant
formally self-dual even codes, DCC 11 (1997), 25–35.211. T. A. Gulliver and M. Harada, Weight enumerators of double circulant codes
and new extremal self-dual codes, DCC 11 (1997), 141–150.212. T. A. Gulliver and M. Harada, Certain self-dual codes over Z4 and the odd
Leech lattice, in Proc. 12th Appl. Alg. Algorithms and Error-Correcting Codes,Lect. Notes Comput. Sci. 1225 (1997), 130–137.
213. T. A. Gulliver and M. Harada, Classification of extremal double circulantself-dual codes of lengths 64 to 72, DCC 13 (1998), 257–269.
214. T. A. Gulliver and M. Harada, On the existence of a formally self-dual even[70, 35, 14] code, Appl. Math. Lett. 11 (1998), 95–98.
215. T. A. Gulliver and M. Harada, Double circulant self-dual codes over Z2k,PGIT 44 (1998), 3105–3123.
216. T. A. Gulliver and M. Harada, New optimal self-dual codes over GF (7), GraphsCombin. 15 (1999) 175–186.
217. T. A. Gulliver and M. Harada, Extremal double circulant Type II code overZ4 and construction of 5 − (24, 10, 36) designs, DM 194 (1999), 129–137.
218. T. A. Gulliver and M. Harada, Double circulant self-dual codes over GF (5),Ars Comb., 56 (2000), 3–13
References 401
219. T. A. Gulliver and M. Harada, On the minimum weight of codes over F5
constructed from certain conference matrices, DCC 31 (2004) 139–145.220. T. A. Gulliver and M. Harada, Classification of extremal double circulant
self-dual codes of lengths 74 to 88, Preprint, 2005.221. T. A. Gulliver, M. Harada and J.-L. Kim, Construction of new extremal self-
dual codes DM 263 (2003), 81–91; erratum 289 (2004), 207.222. T. A. Gulliver and J.-L. Kim, Circulant based extremal additive self-dual
codes over GF (4), PGIT 50 (2004), 359-366.223. T. A. Gulliver, P. R. J. Ostergard and N. I. Senkevitch, Optimal quaternary
linear rate-1/2 codes of length ≤ 18, PGIT 49 (2003), 1540–1543.224. R. C. Gunning, Lectures on Modular Forms, Princton Univ. Press, 1962.225. R. M. Guralnick and P. H. Tiep, Decompositions of small tensor powers and
Larsen’s conjecture, Represent. Theory 9 (2005), 138–208.226. A. J. Hahn and O. T. O’Meara, The Classical Groups and K-Theory, Springer,
1989.227. A. R. Hammons, Jr., P. V. Kumar, A. R. Calderbank, N. J. A. Sloane and P.
Sole, The Z4-linearity of Kerdock, Preparata, Goethals and related codes,PGIT 40 (1994), 301–319 [arXiv: math.CO/0207208].
228. M. Harada, Existence of new extremal doubly-even codes and extremal singly-even codes, DCC 8 (1996), 273–284.
229. M. Harada, The existence of a self-dual [70, 35, 12] code and formally self-dualcodes, Finite Fields Applic. 3 (1997), 131–139.
230. M. Harada, Weighing matrices and self-dual codes, Ars Comb. 47 (1997), 65–73.
231. M. Harada, New extremal ternary self-dual codes, Australas. J. Combin., 17(1998), 133–145.
232. M. Harada, New extremal Type II codes over Z4, DCC 13 (1998), 271–284.233. M. Harada, New 5-designs constructed from the lifted Golay code over Z4, J.
Combin. Designs 6 (1998), 225–229.234. M. Harada, Construction of an extremal self-dual code of length 62, PGIT 45
(1999), 1232–1233.235. M. Harada, New extremal self-dual codes of lengths 36 and 38, PGIT 45 (1999),
2541–2543.236. M. Harada, Self-orthogonal 3-(56, 12, 65) designs and extremal doubly-even
self-dual codes of length 56, DCC, to appear, 2005.237. M. Harada, On the self-dual F5-codes constructed from Hadamard matrices of
order 24, J. Combin. Designs 13 (2005), 152–156.238. M. Harada, T. A. Gulliver and H. Kaneta, Classification of extremal double-
circulant self-dual codes of length up to 62, DM 188 (1998), 127–136.239. M. Harada and H. Kharaghani, Orthogonal designs, self-dual codes and the
Leech lattice, J. Combin. Designs 13 (2005), 184–194.240. M. Harada and H. Kharaghani, Orthogonal designs and MDS self-dual codes,
Australas. J. Combin., to appear, 2005.241. M. Harada and H. Kimura, New extremal doubly-even [64, 33, 12] codes, DCC
6 (1995), 91–96.242. M. Harada and H. Kimura, On extremal self-dual codes, Math. J. Okayama
Univ. 37 (1995), 1–14.243. M. Harada and M. Kitazume, Z4-code constructions for the Niemeier lattices
and their embeddings in the Leech lattice, European J. Combin. 21 (2000),473–485.
402 References
244. M. Harada and M. Kitazume, Z6-code constructions of the Leech lattice andthe Niemeier lattices, European J. Combin. 23 (2002), 573–581.
245. M. Harada, M. Kitazume and A. Munemasa, On a 5-design related to anextremal doubly even self-dual code of length 72, JCT A 107 (2004), 143–146.
246. M. Harada, M. Kitazume, A. Munemasa and B. B. Venkov, On some self-dual codes and unimodular lattices in dimension 48, European J. Combin. 26(2005), 543–557.
247. M. Harada, M. Kitazume and M. Ozeki, Ternary code construction of uni-modular lattices and self-dual codes over Z6, J. Algebraic Combin. 16 (2002),209–223.
248. M. Harada and A. Munemasa, Classification of type IV self-dual Z4-codes oflength 16, Finite Fields Appl. 6 (2000), 244–254.
249. M. Harada and A. Munemasa, A quasi-symmetric 2-(49,9,6) design, J. Com-bin. Des. 10 (2002), 173–179.
250. M. Harada and A. Munemasa, Shadows, neighbors and covering radii of ex-tremal self-dual codes, Preprint, 2005.
251. M. Harada, A. Munemasa and V. D. Tonchev, A characterization of designsrelated to an extremal doubly-even self-dual code of length 48, Ann. Combin.,to appear, 2005.
252. M. Harada and P. R. J. Ostergard, Self-dual and maximal self-orthogonalcodes over F7, DM 256 (2002), 471–477.
253. M. Harada and P. R. J. Ostergard, On the classification of self-dual codes overF5, Graphs and Combinatorics 19 (2003), 203–214.
254. M. Harada and M. Oura, On the Hamming weight enumerators of self-dualcodes over Zk, Finite Fields Appl. 5 (1999), 26–34.
255. M. Harada and M. Ozeki, Extremal self-dual codes with the smallest coveringradius, DM 215 (2000), 271–281.
256. M. Harada, M. Ozeki and K. Tanabe, On the covering radius of ternary ex-tremal self-dual codes, DCC 33 (2004), 149-158.
257. M. Harada, P. Sole and P. Gaborit, Self-dual codes over Z4 and unimodularlattices: a survey, in Algebras and combinatorics (Hong Kong, 1997), ed. K.-P.Shum, E. J. Taft and Z.-X. Wan, Springer, 1999, pp 255–275.
258. M. Harada and V. D. Tonchev, Singly-even self-dual codes and Hadamardmatrices, in Proc. Applied. Alg., Alg. Algorithms and Error-Correcting Codes,ed. G. Cohen, M. Giusti and T. Mora, Lecture Notes Comput. Sci. 948 (1995),279–284.
259. R. H. Hardin and N. J. A. Sloane, New spherical 4-designs, DM 106/107(1992), 255–264. (Topics in Discrete Mathematics, vol. 7, “A Collection ofContributions in Honour of Jack Van Lint”, ed. P. J. Cameron and H. C. A.van Tilborg, North-Holland, 1992.)
260. R. H. Hardin and N. J. A. Sloane, McLaren’s improved snub cube and othernew spherical designs in three dimensions, Discrete Comput. Geometry 15(1996), 429–441.
261. S. Helgason, Differential Geometry, Lie Groups and Symmetric Spaces, Acad-emic Press, NY, 1978.
262. T. Helleseth and P. V. Kumar, Sequences with low correlation, Chapter 21 of[426], pp. 1765–1854.
263. N. Herrmann, Hohere Gewichtszahler von Codes und deren Beziehung zur The-orie der Siegelschen Modulformen, Diplomarbeit, Bonn 1991
References 403
264. R. J. Higgs and J. F. Humphreys, Decoding the ternary Golay code, PGIT39 (1993), 1043–1046.
265. F. Hirzebruch, The ring of Hilbert modular forms for real quadratic fields ofsmall discriminant, in Lect. Notes Math., 627 (1977), 288–323; GesammelteAbhandlungen II (1987), 501–536.
266. G. Hohn, Self-dual codes over the Kleinian four-group, Math. Ann. 327 (2003),227–255 [arXiv: math.CO/0005266].
267. T. Honold and I. Landjev, Linear codes over finite chain rings, Electronic J.Combin. 7 (No. 1, 2000), #11.
268. H. Horimoto and K. Shiromoto, A Singleton bound for linear codes over finitequasi-Frobenius rings, in Applied Algebra, Algebraic Algorithms and Error-Correcting Codes (Honolulu, HI, 1999), Lecture Notes Comput. Sci. 1719(1999), pp. 51–52.
269. S. K. Houghten, C. W. H. Lam, L. H. Thiel and J. A. Parker, The extendedquadratic residue code is the only (48, 24, 12) self-dual doubly-even code, PGIT49 (2003), 53–59.
270. W. C. Huffman, The biweight enumerator of self-orthogonal codes, DM 26(1978), 129–143.
271. W. C. Huffman, Automorphisms of codes with applications to extremal doublyeven codes of length 48, PGIT 28 (1982), 511–521.
272. W. C. Huffman, Decomposing and shortening codes using automorphisms,PGIT 32 (1986), 833–836.
273. W. C. Huffman, On the [24,12,10] quaternary code and binary codes with anautomorphism having two cycles, PGIT 34 (1988), 486–493.
274. W. C. Huffman, On the equivalence of codes and codes with an automorphismhaving two cycles, DM 83 (1990), 265–283.
275. W. C. Huffman, On extremal self-dual quaternary codes of lengths 18 to 28, I,PGIT 36 (1990), 651–660.
276. W. C. Huffman, On 3-elements in monomial automorphism groups of quater-nary codes, PGIT 36 (1990), 660–664.
277. W. C. Huffman, On extremal self-dual quaternary codes of lengths 18 to 28,II, PGIT 37 (1991), 1206–1216.
278. W. C. Huffman, On extremal self-dual ternary codes of lengths 28 to 40, PGIT38 (1992), 1395–1400.
279. W. C. Huffman, On the classification of self-dual codes, Proc. 34th AllertonConf. Commun. Control and Computing, Univ. Ill., Urbana, 1996, pp. 302–311.
280. W. C. Huffman, Characterization of quaternary extremal codes of lengths 18and 20, PGIT 43 (1997), 1613–1616.
281. W. C. Huffman, Decompositions and extremal type II codes over Z4, PGIT 44(1998), 800–809.
282. W. C. Huffman, On the classification and enumeration of self-dual codes, FiniteFields Applic. 11 (2005), 451–490.
283. W. C. Huffman and N. J. A. Sloane, Most primitive groups have messy invari-ants, Advances in Math. 32 (1979), 118–127.
284. W. C. Huffman and V. D. Tonchev, The existence of extremal [50, 25, 10] codesand quasi-symmetric 2-(49,9,6) designs, DCC 6 (1995), 97–106.
285. W. C. Huffman and V. D. Tonchev, The [52, 26, 10] binary self-dual codes withan automorphism of order 7, Finite Fields Appl. 7 (2001), 341–349.
286. W. C. Huffman and V. Y. Yorgov, A [72, 36, 16] doubly even code does nothave an automorphism of order 11, PGIT 33 (1987), 749–752.
404 References
287. J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge Univ.Press, 1990.
288. B. Huppert, Endliche Gruppen I, Springer, 1967.289. J.-I. Igusa, On Siegel modular forms of genus II, Amer. J. Math. 84 (1962),
175–200.290. N. Ito, Symmetry codes over GF (3), JCT A 29 (1980), 251–253.291. N. Jacobson, Basic Algebra II, Freeman, New York, 1980.292. D. B. Jaffe, Binary Linear Codes: New Results on Nonexistence, published
electronically at http://www.math.unl.edu/∼djaffe/codes/code.ps.gz.293. D. B. Jaffe, Optimal binary linear codes of length ≤ 30, DM 223 (2000), 135–
155.294. G. A. Kabatiansky and V. I. Levenshtein, Bounds for packings on a sphere
and in space, Probl. Pered. Inform. 14 (1978), no. 1, 3–25; Problems Inform.Transmission 14 (1978), no. 1, 1–17.
295. S. N. Kapralov and V. D. Tonchev, Extremal doubly-even codes of length 64derived from symmetric designs, DM 83 (1990), 285–289.
296. M. Karlin, New binary coding results by circulants, PGIT 15 (1969), 81–92.297. F. Kasch, Moduln und Ringe, Teubner, Stuttgart, 1977.298. L. S. Kazarin, On certain groups defined by Sidelnikov (in Russian), Mat.
Sb. 189 (No. 7, 1998), 131–144; English translation in Sb. Math. 189 (1998),1087–1100.
299. G. T. Kennedy, Weight distributions of linear codes and the Gleason-Piercetheorem, JCT A 67 (1994), 72–88.
300. I. L. Kheifets, Extension theorem for linear codes over finite quasi-Frobeniusmodules (in Russian), Fundam. Prikl. Mat. 7 (No. 4, 2001), 1227–1236.
301. D. K. Kim, H. K. Kim and J.-L. Kim, Type I codes over GF(4), Preprint,2004.
302. J.-L. Kim, New extremal self-dual codes of lengths 36, 38, and 58, PGIT 47(2001), 386–393.
303. J.-L. Kim, New self-dual codes over GF (4) with the highest known minimumweights, PGIT 47 (2001), 1575–1580.
304. J.-L. Kim and Y. Lee, Euclidean and Hermitian self-dual MDS codes overlarge finite fields, JCT A 105 (2004), 79–95.
305. J.-L. Kim, K. E. Mellinger and V. S. Pless, Projections of binary linear codesonto larger fields, SIAM J. Discrete Math. 16 (2003), 591–603.
306. J.-L. Kim and V. S. Pless, Decoding some doubly-even self-dual [32, 16, 8]codes by hand, in Codes and Designs (Ohio State University, May 2000, theRay-Chaudhuri Festschrift), ed. K. T. Arasu and A. Seress, de Gruyter, Berlin,2002, pp. 165–178.
307. J.-L. Kim and V. S. Pless, Designs in additive codes over GF (4), DCC 30(2003), 187–199.
308. H. Kimura, Extremal doubly even (56, 28, 12) codes and Hadamard matricesof order 28, Australas. J. Combin. 10 (1994), 171–180.
309. O. D. King, The mass of extremal doubly-even self-dual codes of length 40,PGIT 47 (2001), 2558–2560.
310. A. Y. Kitaev, Quantum computations: algorithms and error correction (inRussian), Uspekhi Mat. Nauk. 52 (No. 6, 1997), 53–112; English translationin Russian Math. Surveys 52 (1997), 1191–1249.
311. A. Y. Kitaev, A. H. Shen and M. N. Vyalyi, Classical and Quantum Compu-tation, Amer. Math. Soc., Providence, RI, 2002.
References 405
312. Y. Kitaoka, Arithmetic of Quadratic Forms, Cambridge Univ. Press, 1993.313. M. Kitazume, T. Kondo and I. Miyamoto, Even lattices and doubly even
codes, J. Math. Soc. Japan 43 (1991), 67–87.314. P. B. Kleidman and M. W. Liebeck, The Subgroup Structure of the Finite
Classical Groups, Cambridge Univ. Press, 1988.315. M. Klemm, Selbstduale Codes uber dem Ring der ganzen Zahlen modulo 4,
Arch. Math. (Basel) 53 (1989), 201–207.316. M. Klemm, Eine Invarianzgruppe fur die vollstandige Gewichtsfunktion selb-
stdualer Codes, Arch. Math. (Basel) 53 (1989), 332–336.317. H. Klingen, Introductory Lectures on Siegel Modular Forms , Cambridge Univ.
Press, 1990.318. M. Kneser, Quadratische Formen, Springer, 2002.319. E. Knill, Non-binary unitary error bases and quantum codes, Technical Re-
port LAUR-96-2717, Los Alamos National Laboratory, 1996 [arXiv: quant-ph/9608048].
320. E. Knill, Group representations, error bases and quantum codes, TechnicalReport LAUR-96-2807, Los Alamos National Laboratory, 1996 [arXiv: quant-ph/9608049].
321. E. Knill and R. Laflamme, A theory of quantum error correcting codes, Phys.Rev. A 55 (1997), 900–911 [arXiv: quant-ph/9604034].
322. E. Knill, R. Laflamme and W. Zurek, Resilient quantum computation: Errormodels and thresholds, Proc. R. Soc. Lond. A 454 (1998), 365–384 [arXiv:quant-ph/9702058].
323. M.-A. Knus, Quadratic and Hermitian Forms over Rings, Springer, 1991.324. M.-A. Knus, A. Merkurjev, M. Rost and J.-P. Tignol, The Book of Involutions,
Amer. Math. Soc., Providence, RI, 1998.325. H. Koch, Unimodular lattices and self-dual codes, in Proc. Intern. Congress
Math. Berkeley 1986, Amer. Math. Soc., Providence, RI, Vol. 1, 1987, 457–465.326. H. Koch, On self-dual, doubly even codes of length 32, JCT A 51 (1989),
63–76.327. H. Koch, On self-dual doubly-even extremal codes, DM 83 (1990), 291–300.328. H. Koch, The 48-dimensional analogues of the Leech lattice, Trudy Mat. Inst.
Steklov. 208 (1995), 193–201.329. H. Koch and G. Nebe, Extremal even unimodular lattices of rank 32 and related
codes, Math. Nachr. 161 (1993), 309–319.330. H. Koch and B. B. Venkov, Uber ganzzahlige unimodulare euklidische Gitter,
J. Reine Angew. Math. 398 (1989), 144–168.331. H. Koch and B. B. Venkov, Uber gerade unimodulare Gitter der Dimension
32, III, Math. Nachr. 152 (1991), 191–213.332. W. Kohnen and R. Salvati Manni, Linear relations between theta series, Osaka
J. Math. 41 (2004), 353–356.333. T. Kogiso and K. Tsushima, On an algebra of Siegel modular forms associated
with the theta group Γ2(1, 2), Tsukuba J. Math. 22 (1998), 645–656.334. A. I. Kostrikin and P. H. Tiep, Orthogonal Decompositions and Integral Lat-
tices, de Gruyter, Berlin, 1994.335. I. Krasikov and S. Litsyn, Linear programming bounds for doubly-even self-
dual codes, PGIT 43 (1997), 1238–1244.336. I. Krasikov and S. Litsyn, An improved upper bound on the minimum distance
of doubly-even self-dual codes, PGIT 46 (2000), 274–278.
406 References
337. F. R. Kschischang and S. Pasupathy, Some ternary and quaternary codes andassociated sphere packings, PGIT 38 (1992), 227–246.
338. C. W. H. Lam, The search for a finite projective plane of order 10, Amer.Math. Monthly 98 (1991) 305–318.
339. C. W. H. Lam and V. S. Pless, There is no (24,12,10) self-dual quaternarycode, PGIT 36 (1990), 1153–1156.
340. C. W. H. Lam, L. Thiel and S. Swiercz, The non-existence of finite projectiveplanes of order 10, Canad. J. Math. 41 (1989), 1117–1123.
341. T. Y. Lam, Lectures on Modules and Rings, Springer, 1999.342. T. Y. Lam, A First Course in Noncommutative Rings, Springer, 2nd. ed., 2001.343. S. Lang, Algebra, 3rd. ed., Addison-Wesley, Reading, MA, 1993.344. J. Leech and N. J. A. Sloane, Sphere packing and error-correcting codes, Cana-
dian J. Math., 23 (1971), 718–745. (See also Chapter 5 in [133].)345. J. S. Leon, J. M. Masley and V. S. Pless, Duadic codes, PGIT 30 (1984),
709–714.346. J. S. Leon, V. S. Pless and N. J. A. Sloane, On ternary self-dual codes of
length 24, PGIT 27 (1981), 176–180.347. J. S. Leon, V. S. Pless and N. J. A. Sloane, Self-dual codes over GF(5), JCT
A 32 (1982), 178–194.348. V. I. Levenshtein, Universal bounds for codes and designs, Chapter 6 of [426],
pp. 499–648.349. S. Ling and P. Sole, Type II Codes over F4 +uF4, European J. Combinatorics
22 (2001), 983–997.350. J. H. van Lint, Introduction to Coding Theory, Springer, 1982.351. J. H. van Lint and F. J. MacWilliams, Generalized quadratic residue codes,
PGIT 24 (1978), 730–737.352. S. Litsyn, An updated table of the best binary codes known, Chapter 5 of [426],
pp. 463–498.353. O. Loos, Bimodule-valued Hermitian and quadratic forms, Arch. Math. (Basel)
62 (1994), 134–142.354. X. Ma and L. Zhu, Nonexistence of extremal doubly even self-dual codes,
unpublished manuscript, 1997.355. S. MacLane, Categories for the Working Mathematician, Springer, 2nd ed.,
1998.356. F. J. MacWilliams, A theorem on the distributionm of weights in a systematic
code, Bell Syst. Tech. J. 42 (1963), 79–94.357. F. J. MacWilliams, Orthogonal matrices over finite fields, Amer. Math. Monthly
76 (1969), 152–164.358. F. J. MacWilliams, Orthogonal circulant matrices over finite fields, and how
to find them, JCT 10 (1971), 1–17.359. F. J. MacWilliams, C. L. Mallows and N. J. A. Sloane, Generalizations of
Gleason’s theorem on weight enumerators of self-dual codes, PGIT 18 (1972),794–805.
360. F. J. MacWilliams, A. M. Odlyzko, N. J. A. Sloane and H. N. Ward, Self-dualcodes over GF(4), JCT A 25 (1978), 288–318
361. F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes,North-Holland, Amsterdam, 1977; 11th impression 2003.
362. F. J. MacWilliams, N. J. A. Sloane and J. G. Thompson, Good self-dual codesexist, DM 3 (1972), 153–162.
References 407
363. C. L. Mallows, A. M. Odlyzko and N. J. A. Sloane, Upper bounds for modularforms, lattices and codes, J. Algebra 36 (1975), 68–76.
364. C. L. Mallows, V. S. Pless and N. J. A. Sloane, Self-dual codes over GF (3),SIAM J. Appl. Math. 31 (1976), 649–666.
365. C. L. Mallows and N. J. A. Sloane, An upper bound for self-dual codes, Infor-mation and Control 22 (1973), 188–200.
366. C. L. Mallows and N. J. A. Sloane, Weight enumerators of self-orthogonalcodes, DM 9 (1974), 391–400.
367. C. L. Mallows and N. J. A. Sloane, Weight enumerators of self-orthogonal codesover GF (3), SIAM J. Algebraic and Discrete Methods 2 (1981), 425–460.
368. J. Martinet, Perfect Lattices in Euclidean Spaces, Springer, 2003.369. B. R. McDonald, Finite Rings with Identity, Dekker, NY, 1974.370. R. J. McEliece, E. R. Rodemich, Jr., H. Rumsey and L. R. Welch, New upper
bounds on the rate of a code via the Delsarte-MacWilliams inequalities, PGIT23 (1977), 157–166.
371. J. Milnor and D. Husemoller, Symmetric Bilinear Forms, Springer, 1973.372. T. Miyake, Modular Forms, Springer, 1989.373. T. Molien, Ueber die invarianten der linear Substitutionsgruppe, Sitzungsber
Konig. Akad. Wiss., (1897), 1152–1156.374. E. H. Moore, Double Circulant Codes and Related Algebraic Structures, Ph.D.
Dissertation, Dartmouth College, July 1976.375. D. Mumford, Tata Lectures on Theta I , Birkhauser, Boston, 1983.376. D. Mumford, Tata Lectures on Theta III , Birkhauser, Boston, 1983.377. A. Munemasa, A mass formula for Type II codes over finite fields of char-
acteristic two, in Codes and Designs (Ohio State University, May 2000, theRay-Chaudhuri Festschrift), ed. K. T. Arasu and A. Seress, de Gruyter, Berlin,2002, pp. 207-214.
378. T. Nakayama, On Frobeniusean algebras I, Annals Math. 40 (1939), 611–633.379. G. Nebe, The normaliser action and strongly modular lattices, L’Enseign.
Math. 43 (1997), 67–76.380. G. Nebe, Some cyclo-quaternionic lattices, J. Alg. 199 (1998), 472–498.381. G. Nebe, An analogue of Hecke operators in coding theory, Preprint, 2005
[arXiv: math.NT/0509474].382. G. Nebe, H.-G. Quebbemann, E. M. Rains and N. J. A. Sloane, Complete
weight enumerators of generalized doubly-even self-dual codes, Finite FieldsApplic., 10 2004, 540–550 [arXiv: math.NT/0311289].
383. G. Nebe, E. M. Rains and N. J. A. Sloane, The invariants of the Clifford groups,DCC 24 (2001), 99–121 [arXiv: math.CO/0001038].
384. G. Nebe, E. M. Rains and N. J. A. Sloane, A simple construction for theBarnes-Wall lattices, in Codes, Graphs and Systems: A Celebration of the Lifeand Career of G. David Forney, Jr. on the Occasion of his Sixtieth Birthday,ed. R. E. Blahut and R. Koetter, Kluwer, Boston, 2002, pp. 333–342. [arXiv:math.CO/0207186].
385. G. Nebe, E. M. Rains and N. J. A. Sloane, Codes and invariant theory, Math.Nachr. 274–275 (2004), 104–116. [math.NT/0311046]
386. G. Nebe and B. B. Venkov, Nonexistence of extremal lattices in certain generaof modular lattices, J. Number Theory 60 (2) (1996) 310–317.
387. W. K. Nicholson and M. F. Yousif, Quasi-Frobenius Rings, Cambridge Univ.Press, 2003.
408 References
388. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Infor-mation, Cambridge Univ. Press, 2000.
389. H.-V. Niemeier, Definite quadratische Formen der Dimension 24 und Diskrim-inante 1, J. Number Theory 5 (1973), 142–178.
390. T. Nishimura, A new extremal self-dual code of length 64, PGIT 50 (2004),2173–2174.
391. A. Nobs, Die irreduziblen Darstellungen der Gruppen SL2(Zp), insbesondereSL2(Z2). I. Teil. Comm. Math. Helvetii 51 (1976), 456–489.
392. P. R. J. Ostergard, There exists no Hermitian self-dual quaternary [26, 13, 10]4code, PGIT 50 (2004), 3316–3317.
393. M. Oura, The dimension formula for the ring of code polynomials in genus 4,Osaka J. Math. (1997), 34, pp. 53–72.
394. M. Ozeki, On the basis problem for Siegel modular forms of degree 2, ActaArith. 31 (1976), 17–30.
395. M. Ozeki, Hadamard matrices and doubly even self-dual error-correcting codes,JCT A 44 (1987), 274–287.
396. M. Ozeki, On a class of self-dual ternary codes, Science Reports Hirosaki Univ.36 (1989), 184–191.
397. M. Ozeki, Quinary code construction of the Leech lattice, Nihonkai Math. J.2 (1991), 155–167.
398. M. Ozeki, On the notion of Jacobi polynomials for codes, Math. Proc. Camb.Phil. Soc. 121 (1997), 15–30.
399. M. Ozeki, On covering radius and coset weight distributions of extremal binaryself-dual codes of length 40, Theoret. Comp. Sci. 235 (2000), 283-308.
400. M. Ozeki, On covering radii and coset weight distributions of extremal binaryself-dual codes of length 56, PGIT 46 (2000), 2359-2372.
401. M. Ozeki, Jacobi polynomials for singly even self-dual codes and the coveringradius problems, PGIT 48 (2002), 547–557.
402. M. Ozeki, Notes on the shadow process in self-dual codes, DM 264 (2003),187–200.
403. B. Pareigis, Non-additive ring and module theory I: General theory of monoids,Publ. Math. Debrecen 24 (1977), 189–204.
404. G. Pasquier, Binary self-dual codes construction from self-dual codes over aGalois field F2m , in Combinatorial Mathemetics (Luminy, 1981), ed. C. Bergeet al., Annals Discrete Math. 17 (1983), 519–526.
405. N. J. Patterson, personal communication, 1980.406. A. Peres, Quantum Theory: Concepts and Methods, Kluwer, Dordrecht, 1995.407. L. Ping and K. L. Yeung, Symbol-by-symbol APP decoding of the Golay
code and iterative decoding of concatenated Golay codes, PGIT 45 (1999),2558–2562.
408. P. M. Piret, Algebraic construction of cyclic codes over Z8 with a good Euclid-ean minimum distance, PGIT 41 (1995), 815–817.
409. A. O. Pittenger, An Introduction to Quantum Computing Algorithms,Birkhauser, Boston, 2000.
410. W. Plesken, Lattices of covariant quadratic forms, L’Enseign. Math. 47 (2001),21–56.
411. V. S. Pless, The number of isotropic subspaces in a finite geometry, Rend. Cl.Scienze fisiche, matematiche e naturali, Acc. Naz. Lincei 39 (1965), 418–421.
412. V. S. Pless, On the uniqueness of the Golay codes, JCT 5 (1968), 215–228.
References 409
413. V. S. Pless, On a new family of symmetry codes and related new five-designs,Bull. Amer. Math. Soc. 75 (1969), 1339–1342.
414. V. S. Pless, A classification of self-orthogonal codes over GF (2), DM 3 (1972),209–246.
415. V. S. Pless, Symmetry codes over GF (3) and new five-designs, JCT A 12(1972), 119–142.
416. V. S. Pless, The children of the (32,16) doubly even codes, PGIT 24 (1978),738–746.
417. V. S. Pless, 23 does not divide the order of the group of a (72,36,16) doublyeven code, PGIT 28 (1982), 113–117.
418. V. S. Pless, A decoding scheme for the ternary Golay code, in Proc. 20thAllerton Conf. Comm. Control, Univ. Ill., Urbana, 1982, pp. 682–687.
419. V. S. Pless, On the existence of some extremal self-dual codes, in Enumerationand Design, ed. D. M. Jackson and S. A. Vanstone, Academic Press, NY, 1984,pp. 245–250.
420. V. S. Pless, Q-codes, JCT A 43 (1986), 258–276.421. V. S. Pless, Decoding the Golay codes, PGIT 32 (1986), 561–567.422. V. S. Pless, Extremal codes are homogeneous, PGIT 35 (1989), 1329–1330.423. V. S. Pless, More on the uniqueness of the Golay code, DM 106 (1992), 391-
398.424. V. S. Pless, Parents, children, neighbors and the shadow, Contemporary Math.
168 (1994), 279–290.425. V. S. Pless, Coding constructions, Chapter 2 of [426], pp. 141–176.426. V. S. Pless and W. C. Huffman, eds., Handbook of Coding Theory, Elsevier,
Amsterdam, 2 vols., 1998.427. V. S. Pless, W. C. Huffman and R. A. Brualdi, An introduction to algebraic
codes, Chapter 1 of [426], pp. 3–139.428. V. S. Pless, J. S. Leon and J. E. Fields, All Z4 codes of Type II and length 16
are known, JCT A 78 (1997), 32–50.429. V. S. Pless and J. N. Pierce, Self-dual codes over GF (q) satisfy a modified
Varshamov-Gilbert bound, Information and Control 23 (1973), 35–40.430. V. S. Pless and Z. Qian, Cyclic codes and quadratic residue codes over Z4,
PGIT 42 (1996), 1594–1600.431. V. S. Pless and N. J. A. Sloane, On the classification and enumeration of
self-dual codes, JCT A 18 (1975), 313–335.432. V. S. Pless, N. J. A. Sloane and H. N. Ward, Ternary codes of minimum
weight 6 and the classification of self-dual codes of length 20, PGIT 26 (1980),305–316.
433. V. S. Pless, P. Sole and Z. Qian, Cyclic self-dual Z4-codes, Finite Fields Appl.3 (1997), 48–69.
434. V. S. Pless and J. G. Thompson, 17 does not divide the order of a group of a(72,36,16) code, PGIT 28 (1982), 537–541.
435. V. S. Pless and V. D. Tonchev, Self-dual codes over GF (7), PGIT 33 (1987),723–727.
436. V. S. Pless, V. D. Tonchev and J. S. Leon, On the existence of a certain(64, 32, 12) extremal code, PGIT 39 (1993), 214–215.
437. A. Poli and C. Rigoni, Enumeration of self-dual 2k circulant codes, in Ap-plied Algebra, Algorithmics and Error-Correcting Codes (Toulouse, 1984), Lect.Notes. Comput. Sci. 228 (1986), 61–70.
410 References
438. H.-G. Quebbemann, On even codes, DM 98 (1991), 29–34.439. H.-G. Quebbemann, Modular lattices in euclidean spaces, J. Number Th. 54
(1995), 190–202.440. H.-G. Quebbemann, Atkin-Lehner eigenforms and strongly modular lattices.
L’Ens. Math. 43 (1997), 55–65.441. H.-G. Quebbemann and E. M. Rains, On the involutions fixing the class of a
lattice, J. Number Theory, 101 (2003), 185–194.442. E. M. Rains, Shadow bounds for self-dual codes, PGIT 44 (1998), 134–139.443. E. M. Rains, Quantum weight enumerators, PGIT 44 (1998), 1388–1394 [arXiv:
quant-ph/9612015].444. E. M. Rains, Quantum codes of minimum distance two, PGIT 45 (1999), 266–
271 [arXiv: quant-ph/9704043].445. E. M. Rains, Nonbinary quantum codes, PGIT 45 (1999), 1827–1832 [arXiv:
quant-ph/9703048].446. E. M. Rains, Optimal self-dual codes over Z4, DM 203 (1999), 215–228.447. E. M. Rains, Quantum shadow enumerators, PGIT 45 (1999), 2361–2366
[arXiv: quant-ph/9611001].448. E. M. Rains, Monotonicity of the quantum linear programming bound, PGIT
45 (1999), 2489–2492 [arXiv: quant-ph/9802070].449. E. M. Rains, Polynomial invariants of quantum codes, PGIT 46 (2000), 54–59
[arXiv: quant-ph/9704042].450. E. M. Rains, Bounds for self-dual codes over Z4, Finite Fields Appl. 6 (2000),
146–163.451. E. M. Rains, New asymptotic bounds for self-dual codes and lattices, PGIT
49 (2003), 1261–1274 [arXiv: math.CO/0104145].452. E. M. Rains, R. H. Hardin, P. W. Shor and N. J. A. Sloane, A nonadditive
quantum code, Phys. Rev. Lett. 79 (1997), 953–954 [arXiv: quant-ph/9802061].453. E. M. Rains and N. J. A. Sloane, The shadow theory of modular and unimodu-
lar lattices, J. Number Theory, 73 (1998), 359–389 [arXiv: math.CO/0207294].454. E. M. Rains and N. J. A. Sloane, Self-dual codes, Chapter 1 of [426], pp.
177-294 [arXiv: math.CO/0208001].455. D. K. Ray-Chaudhuri, Some results on quadrics in finite projective geometry
based on Galois fields, Canad. J. Math. 14 (1962), 129–138.456. I. S. Reed, X. Yin and T. K. Truong, Algebraic decoding of the (32, 16, 8)
quadratic residue code, PGIT 36 (1990), 876–880.457. I. Reiner, Maximal Orders, Academic Press, NY, 1975.458. B. Reznick, Some constructions of spherical 5-designs, Linear Algebra and Its
Applications, 226/228 (1995), 163–196.459. J. Rifa, A new algebraic algorithm to decode the ternary Golay code, Inform.
Process. Lett. 68 (1998), no. 6, 271–274.460. L. H. Rowen, Ring Theory, Academic Press, San Diego, Student ed., 1991.461. B. Runge, On Siegel modular forms I, J. Reine Angew. Math. 436 (1993),
57–85.462. B. Runge, On Siegel modular forms II, Nagoya Math. J. 138 (1995), 179–197.463. B. Runge, The Schottky ideal, in Abelian Varieties (Egloffstein, 1993), de
Gruyter, Berlin, 1995, pp. 251–272.464. B. Runge, Codes and Siegel modular forms, DM 148 (1996), 175–204.465. R. P. Ruseva, Uniqueness of the [36, 18, 8] double circulant code, in Proc. In-
ternat. Workshop on Optimal Codes and Related Topics, May 26–June 1, 1995,Sozopol, Bulgaria, 126–129.
References 411
466. R. P. Ruseva, New extremal self-dual codes of length 36, in Proc. of the 25th
Spring Conf. of the Union of Bulgarian Mathematicians, Bulgarian AcademySci., 1996, 150–153.
467. R. P. Ruseva, Self-dual [24, 12, 8] codes with a non-trivial automorphism oforder 3, Finite Fields Appl. 8 (2002), 34–51.
468. R. A. Sack, Interpretation of Lagrange’s expansion and its generalization toseveral variables as integration formulas, J. SIAM 13 (1965), 47–59.
469. R. A. Sack, Generalization of Lagrange’s expansion for functions of severalimplicitly defined variables, J. SIAM 13 (1965), 913–926.
470. R. A. Sack, Factorization of Lagrange’s expansion by means of exponentialgenerating functions, J. SIAM 14 (1966), 1–15.
471. C. H. Sah, Cohomology of split group extensions: II, J. Algebra 45 (1977),17–68.
472. A. Samorodnitsky, On linear programming bounds for spherical codes anddesigns, Discrete Comput. Geom., 31 (2004), 385–394.
473. R. Scharlau and R. Schulze-Pillot, Extremal lattices, in Algorithmic Algebraand Number Theory, ed. B. H. Matzat, G. M. Greuel and G. Hiss, Springer,1999, pp. 139–170.
474. W. Scharlau and D. Schomaker, personal communication, April 1991.475. W. Scharlau, Quadratic and Hermitian Forms, Springer, 1985.476. P. Schmid, On the automorphism group of extraspecial 2-groups, J. Algebra
234 (2000), 492–506.477. B. Schoeneberg, Elliptic Modular Functions, Springer, 1974.478. B. Segre, Le geometrie di Galois, Ann. Mat. Pura Appl. (4) 48 (1959), 1–96.479. J.-P. Serre, Linear Representations of Finite Groups, Springer, 1977.480. J.-P. Serre, Cours d’arithmetique, Presses Universitaires de France, 3rd. ed.,
Paris, 1988. English translation of 1st edition published by Springer, 1977.481. P. D. Seymour and T. Zaslavsky, Averaging sets: a generalization of mean
values and spherical designs, Advances in Math., 52 (1984), 213–240.482. G. C. Shephard and J. A. Todd, Finite unitary reflection groups, Canad. J.
Math. 6 (1954), 274–304.483. G. Shimura, On modular forms of half integral weight, Annals Math. 97 (1973),
440–481.484. D.-J. Shin, P. V. Kumar and T. Helleseth, 5-designs from the lifted Golay
code over Z4 via an Assmus-Mattson type approach, DM 241 (2001), 479–487.485. K. Shiromoto and L. Storme, A Griesmer bound for linear codes over
finite quasi-Frobenius rings, Discrete Applied Math. 128 (2003), 263–274[cage.rug.ac.be/∼ls/artgriesmerwcc2001-35final.pdf].
486. P. W. Shor, “Fault-tolerant quantum computation,” Proc. 37th Sympos. Foun-dations of Computer Science, IEEE Computer Society Press, 1996, pp. 56–65[arXiv: quant-ph/9605011].
487. P. W. Shor and R. Laflamme, Quantum analog of the MacWilliams identitiesfor classical coding theory, Phys. Rev. Lett. 78 (1997), 1600–1602 [arXiv: quant-ph/9610040].
488. P. W. Shor and N. J. A. Sloane, A family of optimal packings in Grassmannianmanifolds, J. Algebraic Combin. 7 (1998), 157–163 [arXiv: math.CO/0208003].
489. I. Siap, Linear codes over F2 + uF2 and their complete weight enumerators,in Codes and Designs (Ohio State University, May 2000, the Ray-ChaudhuriFestschrift), ed. K. T. Arasu and A. Seress, de Gruyter, Berlin, 2002, pp. 259–271.
412 References
490. V. M. Sidelnikov, On a finite group of matrices and codes on the Euclideansphere (in Russian), Probl. Pered. Inform. 33 (1997), 35–54 (1997); Englishtranslation in Problems Inform. Transmission 33 (1997), 29–44 .
491. V. M. Sidelnikov, On a finite group of matrices generating orbit codes onthe Euclidean sphere, in Proceedings IEEE Internat. Sympos. Inform. Theory,Ulm, 1997, IEEE Press, 1997, p. 436.
492. V. M. Sidelnikov, Spherical 7-designs in 2n-dimensional Euclidean space, J.Algebraic Combin. 10 (1999), 279–288.
493. V. M. Sidelnikov, Orbital spherical 11-designs in which the initial point is aroot of an invariant polynomial (in Russian), Algebra i Analiz 11 (No. 4, 1999),183–203.
494. C. L. Siegel, Einfurung in die Theorie der Modulfunktionen n-ten Grades,Math. Ann. 116 (1939), 617–657; Gesammelte Abhandlungen II (1966), pp.97–137.
495. C. L. Siegel, Berechnung von Zetafunktionen an ganzzahligen Stellen, Nachr.Akad. Wiss. Gottingen 10 (1969), 87–102.
496. J. Simonis, The [18, 9, 6] code is unique, DM 106 (1992), 439–448.497. N. P. Skoruppa, MODI: A modular forms dimension calculator, published elec-
tronically at wotan.algebra.math.uni-siegen.de/∼modi/, 2005.498. N. J. A. Sloane, Is there a (72, 36) d = 16 self-dual code?, PGIT 19 (1973),
251.499. N. J. A. Sloane, Weight enumerators of codes, in Combinatorics, ed. M. Hall Jr.
and J. H. van Lint, Mathematical Centre, Amsterdam and Reidel PublishingCo., Dordrecht, Holland, 1975, pp. 115–142.
500. N. J. A. Sloane, Error-correcting codes and invariant theory: New applicationsof a nineteenth-century technique, Amer. Math. Monthly 84 (1977), 82–107.
501. N. J. A. Sloane, Binary codes, lattices and sphere-packings, in CombinatorialSurveys: Proceedings of the Sixth British Combinatorial Conference, ed. P. J.Cameron, Academic Press, NY, 1977, pp. 117–164.
502. N. J. A. Sloane, Codes over GF (4) and complex lattices, J. Algebra 52 (1978),168–181.
503. N. J. A. Sloane, Self-dual codes and lattices, in Relations Between Combina-torics and Other Parts of Mathematics, Proc. Symp. Pure Math., Vol 34, Amer.Math. Soc., Providence, RI, 1979, pp. 273–308.
504. N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences, publishedelectronically at www.research.att.com/∼njas/sequences/, 2005.
505. N. J. A. Sloane, R. H. Hardin and P. Cara, Spherical Designs in Four Di-mensions (Extended Abstract) in Proceedings Information Theory Workshop(Paris, April 2003), IEEE Press, 2003, pp. 253–257.
506. L. Smith, Polynomial Invariants of Finite Groups, Peters, Wellesley, MA, 1995.507. S. L. Snover, The Uniqueness of the Nordstrom-Robinson and the Golay Binary
Codes, Ph.D. Dissertation, Department of Mathematics, Michigan State Univ.,1973.
508. S. L. Sobolev, Cubature formulae on the sphere invariant under finite groups ofrotations (Russian), Doklady Akademii Nauk SSR, 146 (No. 2, 1962), 310–313;translation in Soviet Mathematics Doklady, 3 (1962), 1307–1310.
509. G. Solomon, Golay encoding/decoding via BCH-Hamming, Comput. Math.Appl., 39 (2000), 103–108.
510. E. Spence and V. D. Tonchev, Extremal self-dual codes from symmetric de-signs, DM 110 (1992), 265–268.
References 413
511. R. P. Stanley, Relative invariants of finite groups generated by pseudoreflec-tions, J. Algebra 49 (1977), 134–148.
512. R. P. Stanley, Invariants of finite groups and their application to combinatorics,Bull. Amer. Math. Soc. 1 (1979), 475–511.
513. A. M. Steane, Multiple particle interference and quantum error correction,Proc. Roy. Soc. London A, 452 (1996), 2551–2577 [arXiv: quant-ph/9601029].
514. A. M. Steane, Simple quantum error correcting codes, Phys. Rev. Lett. 77(1996), 793–797 [arXiv: quant-ph/9605021].
515. J. Stolze and D. Suter, Quantum Computing: A Short Course from Theory toExperiment, Wiley-VCH, Weinheim, Germany, 2004.
516. B. Sturmfels, Algorithms in Invariant Theory, Springer, 1993.517. K. Tanabe, An Assmus-Mattson theorem for Z4-codes, PGIT 46 (2000), 48–53.518. K. Tanabe, A new proof of the Assmus-Mattson theorem for non-binary codes,
DCC 22 (2001), 149–155.519. H. Tapia-Recillas and G. Vega, On Z2k -linear and quaternary codes, SIAM J.
Discrete Math. 17 (2003), 103–113.520. A. Terras, Fourier Analysis on Finite Groups and Applications, Cambridge
Univ. Press, 1999.521. J. G. Thompson, Weighted averages associated to some codes, Scripta Math.
29 (1973), 449–452.522. V. D. Tonchev, Self-orthogonal designs and extremal doubly-even codes, JCT
A 52 (1989), 197–205.523. V. D. Tonchev, Self-dual codes and Hadamard matrices, Discr. Appl. Math.
33 (1991), 235–240.524. V. D. Tonchev, Codes and designs, Chapter 15 of [426], pp. 1229–1268.525. V. D. Tonchev and R. V. Raev, Cyclic 2-(17, 8, 7) designs and related doubly
even codes, Comp. Rend. Acad. Bulg. Sci. 35 (1982), 1367–1370.526. V. D. Tonchev and V. Y. Yorgov, The existence of certain extremal [54, 27, 10]
self-dual codes, PGIT 42 (1996), 1628–1631.527. H.-P. Tsai, Existence of some extremal self-dual codes, PGIT 38 (1992), 1829–
1833.528. H.-P. Tsai, The covering radius of extremal self-dual code D11 and its appli-
cation, PGIT 43 (1997), 316–319.529. H.-P. Tsai, Extremal self-dual codes of lengths 66 and 68, PGIT 45 (1999),
2129–2133.530. H.-P. Tsai and Y.-J. Jiang, Some new extremal self-dual [58,29,10] codes,
PGIT 44 (1998), 813–814.531. J. V. Uspensky, Theory of Equations, McGraw-Hill, NY, 1948.532. A. Vardy, The Nordstrom-Robinson code: representation over GF(4) and effi-
cient decoding, PGIT 40 (1994), 1686–1693.533. B. B. Venkov, The classification of integral even unimodular 24-dimensional
quadratic forms, Trudy Matemat. Inst. Steklova 148, 65–76; Proc. Steklov Inst.Math. (1980), 63–74; [133, Chap. 18].
534. B. B. Venkov, Reseaux et designs spheriques, in Reseaux euclidiens, designsspheriques et formes modulaires, ed. J. Martinet, Monogr. Enseign. Math., 37,Geneva, 2001, pp. 10–86.
535. M. Ventou and C. Rigoni, Self-dual doubly circulant codes, DM 56 (1985),291–298.
536. G. E. Wall, On Clifford collineation, transform and similarity groups IV: anapplication to quadratic forms, Nagoya Math. J. 21 (1962), 199–222.
414 References
537. Z.-X. Wan, Studies in finite geometries and the construction of incompleteblock designs, I: Some “Anzahl” theorems in symplectic geometry over finitefields (Chinese), Acta Math. Sinica 15 354-361; Chinese Math.–Acta 7 (1965)55-62.
538. Z.-X. Wan, Geometry of Classical Groups Over Finite Fields, Studentlitteratur,Lund; Chartwell-Bratt Ltd., Bromley, 1993.
539. Z.-X. Wan, Quaternary Codes, World Scientific, Singapore, 1997.540. H. N. Ward, A restriction on the weight enumerator of self-dual codes, JCT
21 (1976), 253–255.541. H. N. Ward, Divisible codes, Arch. Math. (Basel) 36 (1981), 485–494.542. H. N. Ward, personal communication.543. H. N. Ward, A bound for divisible codes, PGIT 38 (1992), 191–194.544. H. N. Ward, Quadratic residue codes and divisibilty, Chapter 9 of [426], pp.
827–870.545. H. N. Ward and J. A. Wood, Characters and the equivalence of codes, JCT
A 73 (1996), 348–352.546. A. Weil, Sur certaines groupes d’operateurs unitaires, Acta Math. 111 (1964),
143–211; Oeuvres Scientifiques III, Springer, 1979, pp. 1–69.547. E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge
Univ. Press, 4th ed., 1963.548. D. L. Winter, The automorphism group of an extraspecial p-group, Rocky Mtn.
J. Math. 2 (1972), 159–168.549. J. Wolfmann, New decoding methods [for] the Golay code (24, 12, 8), in Combi-
natorial mathematics (Marseille-Luminy, 1981), North-Holland, Amsterdam,1983, pp. 651–656.
550. J. Wolfmann, Nouvelles methodes de decodage du code de Golay (24, 12, 8),Rev. CETHEDEC no. 2, (1982), 79-88.
551. J. Wolfmann, A class of doubly even self-dual binary codes, DM 56 (1985),299–303.
552. J. A. Wood, Extension theorems for linear codes over finite rings, in AppliedAlgebra, Algebraic Algorithms and Error-Correcting Codes, Proc. 12th Inter-nat. Symp., AAECC-12, Toulouse, June, 1997, ed. T. Mora and H. Mattson,Lect. Notes Comput. Sci. 1255 (1997), pp. 329–340.
553. J. A. Wood, Weight functions and the extension theorem for linear codes overfinite rings, in Finite Fields: Theory, Applications and Algorithms, Proc. FourthInternat. Conf. Finite Fields, Waterloo, August 1997, ed. R. C. Mullin and G.L. Mullen, Contemp. Math. 225, Amer. Math. Soc., Providence, RI, 1999, pp.231–243.
554. J. A. Wood, Duality for modules over finite rings and applications to codingtheory, Amer. J. Math. 121 (1999), 555–575.
555. V. Y. Yorgov, Binary self-dual codes with automorphisms of odd order, Probl.Pered. Inform. 19 (1983); English translation in Prob. Inform. Trans. 19(1983), 11–24.
556. V. Y. Yorgov, A method for constructing inequivalent self-dual codes withapplications to length 56, PGIT 33 (1987), 77–82.
557. V. Y. Yorgov, Doubly-even codes of length 64, Probl. Pered. Inform. 22 (1986),35–42; English translation in Prob. Inform. Trans. 22 (1986), 277–284.
558. V. Y. Yorgov, The extremal codes of length 42 with an automorphism of order7, DM 190 (1998), 210–213.
References 415
559. V. Y. Yorgov, On the minimal weight of some singly-even codes, PGIT 45(1999), 2539–2541.
560. V. Y. Yorgov, New self-dual codes of length 106, Congr. Numer. 162 (2003),111-117.
561. V. Y. Yorgov and R. P. Ruseva, Two extremal codes of length 42 and 44,Probl. Pered. Inform. 29 (1993), 99–103; English translation in Prob. Inform.Trans. 29 (1994), 385–388.
562. V. Y. Yorgov and N. Yankov, On the extremal binary codes of lengths 36and 38 with an automorphism of order 5, Proc. of the 5th International Work-shop on Algebraic and Combinatorial Coding Theory, June 1–7, 1996, Sozopol,Bulgaria, 307–312.
563. V. Y. Yorgov and N. P. Ziapkov, Doubly-even self-dual [40, 20, 8] codes withan automorphism of odd order, Probl. Pered. Inform. 32 (1996), 41–46; Englishtranslation in Prob. Inform. Trans. 32 (1996), 253–257.
564. J. Yuan and C.-M. Leung, Two-level decoding of the (32, 16, 8) quadraticresidue code, Southeast Asian Bull. Math. 18 (1994), 173–182.
565. H. Zassenhaus, Uber eine Verallgemeinerung des Henselschen Lemmas, Arch.Math. (Basel) V (1954), 317–325.
566. S. Zhang, On the nonexistence of extremal self-dual codes, Discrete Appl. Math.91 (1999), 277–286.
567. S. Zhang and S. Li, Some new extremal self-dual codes with lengths 42, 44,52, and 58, DM 228 (2001), 147–150.
Index
+, gluing notation 280{{ }} , structure map 12⊥ 4, 6, 85[−1] map 14, 104[N, k, d]q code 30[r] map 42I 412II 413 454E 424H 474H+ 484H+ (additive over F4) 404H+II (Type II additive over F4) 40
4Z 52
Aaltonen, M. 327, 379, 382, 391Alekseevskii, A. V. vii, 391alphabet 2Amrani, O. xii, 391Anderson, J. B. 1, 391Anderson, J. L. xii, 391Andrianov, A. N. 267, 391anti-automorphism 7Aoki, T. 391Arasu, K. T. 345, 391Araya, M. 345, 392Ashikhmin, A. 379, 381, 382, 392Assmus, E. F., Jr. 80, 392Atkin, A. O. L. 30, 44, 66, 322, 336,
339, 349, 363, 392, 399Atkin-Lehner involution 259Aut(C) 24
automorphism groupof code 24of representation 22of Type 165quantum 373strict 24weak, of code 24weak, of representation 23
Baartmans, A. 336, 392Bachoc, C. x, 40, 81, 162, 197, 221, 234,
244, 247, 326, 341, 345, 392, 394Bajnok, B. 162, 392Bak, A. 13, 14, 392Bannai, E. x, 40, 81, 184, 392Barnes, E. S. x, 171, 392Bartels, H.-J. 190, 392Barton, D. E. 62, 397Bayer-Fluckiger, E. 270Beenker, G. F. M. 334, 338, 392Beery, Y. xii, 391Bennett, C. H. 172, 392Benson, D. J. 155, 157, 159, 392Berlekamp, E. R. v, 392β 7β-determinant 277Betsumiya, K. x, xii, xiii, 197, 222, 225,
244, 339, 345, 349, 363, 393Bhargava, V. K. 332, 334, 336, 393,
400bicomplete abelian category 105bilinear
form 2, 3, 83, 84, 103equivalent 84
418 Index
nonsingular 6similar 84weakly equivalent 84
Bilous, R. T. 332, 351, 393Blake, I. F. 337, 393Blaum, M. xii, 393van der Blij, F. 260, 393Bocherer, S. 301, 393Bolt, B. vii, x, 171, 393, 394Bonnecaze, A. x, 40, 81, 234, 326, 345,
394Borcherds, R. E. 301, 394Bosma, W. 394Bossert, M. xii, 394Boukliev, I. 332bound
Bonnecaze, Sole, Bachoc andMourrain 323
Conway–Sloane 320Gilbert–Varshamov 328, 330integer programming 315Krasikov–Litsyn 326linear programming 315, 316lower 328Mallows–Sloane 319Rains 320Singleton 322, 381sphere-packing 381upper 314
Bouyukliev, I. 367, 394Bouyuklieva, S. 332, 367, 394Braun, H. 260, 395Broue, M. vii, 172, 395Brouwer, A. E. 313, 350, 390, 395Brualdi, R. A. 1, 25, 326, 328, 332, 334,
358, 395, 409Bruck, J. xii, 393Bruinier, J. H. 301, 302, 395Brylinski, R. K. 370, 396Bussemaker, F. C. 332, 395Buyuklieva, S. 332
C(ρ), Clifford-Weil group 139Cm(ρ), Clifford-Weil group of genus m
141Calderbank, A. R. viii, x–xii, 1, 26, 70,
73, 77, 172, 173, 234, 341, 345, 365,369, 372, 375, 379, 385–387, 394,395, 401
Calderbank-Shor-Steane construction197, 372
Cameron, P. J. 172, 395Cannon, J. 394, 395Cara, P. 162, 412Carlach, J.-C. 395Carlet, C. 73, 396Cassels, J. W. S. 236, 252, 396category 103
closed 109of quadratic groups 105
center set 362central product xivchain ring v, 151Chang, I. L. 372Chapman, R. 396character group 35Chen, G. 370, 396Cheng, Y. 337, 340, 396Chevalley, C. 157, 396Choie, Y.-J. x, xiii, 222, 225, 393, 394,
396Chuang, I. L. 370, 408Cleve, R. 382, 396Clifford
group 171Clifford group viiClifford-Weil group vi, 129, 139, 141,
142genus-m 141universal 146
co-unitary group 131code 2, 4, 85
additive 42, 375additive Hermitian 26binary 40closed 83, 86component 280double circulant 333doubly-even 16, 40, 41doubly-even, generalized 44dual 4, 6, 83, 85, 95equivalent 23extremal xi, 314, 324formally self-dual xii, 80, 167glue 99Hermitian 47hexacode 223in representation ρ 6
Index 419
isodual xiiisotropic x, 4Kerdock 1Lee-extremal 325linear v, 5, 42, 47linear over Fq 68minimal distance of 371Nordstrom-Robinson xii, 1, 73norm extremal 324not closed 86optimal xi, 313, 325over commutative ring 88over quasi-Frobenius ring 89p-adic 60permutation-equivalent 23Pless symmetry 337Preparata 1pure 371quadratic residue 63–65, 67–69, 333
Euclidean 65Hermitian 69
quantum 371additive 374
quasi-cyclic 386quaternary 42quaternary additive 48quaternary linear 47Reed-Muller 353Reed-Solomon 64repetition 61, 173self-dual 4, 6, 371self-dual over Fp 16self-orthogonal 4, 6, 285, 367singly-even 16, 40, 41stabilizer 375ternary 45Type I 40, 41Type II 40, 41Type III 67Type of 15weakly equivalent 23Z/4Z-linear 52
codeword 2cogenerator 88Cohen, A. M. 350, 395Cohen, H. 257, 396complete weight enumerator 30component code 280composition 29
Construction A vii, 197, 263, 264, 270,300, 301, 303
Conway, J. H. x, xii, 24–26, 42, 69, 70,72, 74, 172, 221, 260, 280, 283, 320,332, 334, 336, 339–342, 351, 353,354, 360, 362, 363, 366, 396, 397
Coppersmith, D. 338, 397Coxeter, H. S. M. 209, 397Craig, M. 270cwe 30cwem 33cyclic group Zn xv
(d10e7f1)+ 179, 221, 281, 333
d+12 179, 221, 333
dN 63, 351Dai, Z.-D. 349, 398Dalan, D. B. 332, 397Danielsen, L. E. 339, 365, 397David, F. N. 62, 397Dawson, E. 338, 397decoding xiiDelsarte, P. 161, 162, 315, 397Derksen, H. 155, 397Desidiri Bracco, A. 394dimension 30distance
Hamming 29Lee 30
DiVincenzo, D. 172, 392Dodunekov, S. M. xii, 397Dontcheva, R. 332, 336, 397double circulant code 334doubly-even code 5, 16, 40, 41Dougherty, S. T. x, 40, 70, 81, 184,
324, 332, 336, 342, 345, 358, 392,394, 396, 398
dual xvcode 4, 83, 85, 95in a representation 6lattice 252, 277subgroup 35, 85
Duke, W. x, 172, 183, 264, 398Duursma, I. M. 366, 398Dym, H. 37, 398
E(V ), Heisenberg group 140e7 63, 281, 352, 366, 372
420 Index
e8 v–vii, 61, 63, 73, 173, 178, 183, 184,221, 333, 350, 352, 353, 365
Ebeling, W. 252, 253, 263, 270, 398Eholzer, W. 304, 306, 398Eichler, M. 261, 275, 398Eisenstein series 250, 327, 331Elkies, N. D. vii, 326, 398Enguehard, M. vii, 172, 395enumeration
binary self-dual codes 350ε 6equivalence 84
permutation 23weak 84
equivalent 374bilinear forms 84codes 23globally 373locally 373
erasure 371Esmaeili, M. xii, 398Euclidean norm 30Euclidean-extremal 317even 22
lattice 252level 253, 255map 3matrix 22, 229sublattice 260
Evn 22excess 256extraspecial group 140extremal
code 314shadow- 326
extremal code xi, 313extremal weight enumerator 314, 316,
318
Fq, field xivFaith, C. 89, 398faithful 14families
examples of 60list of 40, 78
Fekri, F. xii, 398Feng, X.-N. 349, 398Fields, J. E. 70, 342, 366, 398, 399, 409form
bilinear 2, 3, 83, 84, 103Hermitian 94Jacobi 261subquotient 98
form algebra 272form ideal 14form k-algebra 116form order 276form parameter 13form ring 13
automorphism of 23finite 13matrix 19quasisimple 193quotient of 15radical of 15representation 13semisimple 14simple 14, 193split type 195sub- 17, 276, 317triangular 18
Forney, G. D., Jr. 73, 399free component 281free functor 106Freitag, E. vii, 229, 261, 263, 264, 279,
399Frohlich, A. 189, 399full weight enumerator 31functor
free 106squaring Q 106underlying group 106
fwe 31fwem 34
g12 67, 209, 210, 293, 361g23 63g24 v, vii, 63, 74, 179, 183, 184, 333, 353Gaborit, P. x, xii, xiii, 30, 44, 66, 70,
71, 234, 244, 322, 326, 331, 332,334, 336, 338, 339, 341, 342, 345,349, 350, 363, 366, 391, 392, 394,396, 398, 399, 402
Γ0(�) 253γρ(φ), Gauss sum 145Gaulter, M. 324, 399Gauss sum 144, 145generated by 85
Index 421
generatorfor code 60matrix 252module 116
genus-m ρ-symmetrized weightenumerator 33
genus-m full weight enumerator 34geometry
orthogonal 350symplectic 350unitary 350
Georgiou, S. 345, 393, 399GLn(Fq),general linear group xivGlasby, S. P. 172, 399Gleason, A. M. v, 31, 80, 183, 209, 399global conjugation 43glue
code 99word 280
gluing theory 280, 282Goethals, J.-M. 161, 162, 397, 399Golay code
2-adic 773-adic 77half- 353odd 353over Z/4Z 74quarter- 354
good polynomial basis 155good self-dual codes exist 329Good, I. J. 318, 400Gordon, D. M. xii, 400Gottesman, D. 375, 381, 386, 400GR(q, f), Galois ring 51Graeffe’s method 74Gram matrix 252Grassl, M. 385, 390, 400Gray map 66, 73Greferath, M. xii, 73, 366, 398, 400Griess, R. L. 172, 400Griffiths, P. 265, 400Gross, B. vii, 400ground ring 2group
alternating An 364automorphism 24character 35Clifford vii, 171co-unitary 131
cyclic Zn xvextraspecial 140Heisenberg 140hyperbolic co-unitary 133, 136, 273icosahedral 212Mathieu 63, 67metaplectic viiof equivalences 23of weak equivalences 23p-Clifford 176parabolic 130permutation 24quadratic 104sextet 354symmetric SN xv, 42symplectic 262theta 229, 249, 263, 279weak automorphism 24Witt 103, 123, 287Witt, projective 122
group ring 31Gunther, A. 246Gulliver, T. A. x, xii, 40, 81, 197, 326,
332, 334, 336, 339, 345, 358, 367,386, 391, 393, 394, 398, 400, 401
Gunning, R. C. 401Guralnick, R. M. 191, 401
H# 35h5 69, 298h6 68, 76, 224, 364, 386Hadamard matrix 362Hahn, A. J. viii, 13, 131, 194, 401half-space 275Hall, P. 208Hamming
distance 29weight 29weight enumerator 30
quantum 376Hamming code
2-adic 77quantum 380, 387
Hamming-extremal 316Hammons, A. R., Jr. viii, xii, 1, 70, 73,
395, 401Harada, M. x, xii, xiii, 40, 70, 81, 184,
197, 234, 324, 326, 332, 334, 336,
422 Index
338, 339, 342, 345, 358, 363, 367,391–394, 397–402
Hardin, R. H. 162, 172, 173, 376, 380,395, 396, 402, 410, 412
Harris, J. 265, 400heat equation 269Helgason, S. 273, 402Helleseth, T. 51, 326, 402, 411Hermitian
form 94inner product 36
Herrmann, N. 263, 402hexacode 68Higgs, R. J. xii, 403Hilbert half-plane 269Hilbert theta series 270Hironaka decomposition 156Hirzebruch, F. 270, 403Hohn, G. 69, 341, 349, 365, 403Honold, T. 151, 403Horimoto, H. 89, 403Houghten, S. K. 332, 335, 403Huffman, W. C. x, xi, 1, 70, 157, 183,
313, 326, 328, 331, 332, 334, 335,337, 339–342, 345, 399, 403, 409
Humphreys, J. E. 157, 404Humphreys, J. F. xii, 403Huppert, B. 208, 404Husemoller, D. 252, 260, 407hwe 30hyperbolic co-unitary group 133, 136,
273generators 136
i2 vi, 61, 68, 173, 178, 183, 192ideal in twisted ring 9idempotent 11, 18
symmetric 101, 136Igusa, J.-I. 264, 404injective 87inner product
Euclidean 42, 43Hermitian 36, 47standard 40trace 48
integersmod n xvn-adic xv
internal hom IHom 108
invariantbasic 155primary 155relative 159ring 155secondary 156
involutionAtkin-Lehmer 259
isometry 115isomorphism 114
weak 114isotropic
code 4lattice 277subspace 350
Ito, N. 337, 404
J , anti-automorphism 7Jacobi form 261Jacobi identity vii, 251, 253, 266Jacobi-Siegel theta series 265Jacobson, N. 194, 404Jaffe, D. B. 35, 382, 404Jeong, E. 396Jiang, Y.-J. 332, 413
K-theory viiiKabatiansky, G. A. 328, 404Kaneta, H. 332, 401Kantor, W. M. x, 172, 387, 395Kapralov, S. N. 332, 404Karlin, M. 334, 404Kasch, F. 89, 404Kazarin, L. S. 172, 404Kemper, G. 155, 397Kendall, M. G. 62, 397Kennedy, G. T. 80, 404Khandani, A. K. xii, 398Kharaghani, H. 345, 401Kheifets, I. L. 89, 404Kim, D. K. 339, 363, 404Kim, H. 396Kim, H. K. 339, 363, 404Kim, J.-L. x, xii, 326, 331, 332, 336,
339, 341, 345, 363, 386, 399, 401,404
Kim, N. 396Kimura, H. 332, 401, 404King, O. D. 332, 351, 404
Index 423
Kitaev, A. Y. 172, 370, 404Kitaoka, Y. 125, 252, 257, 405Kitazume, M. 326, 332, 345, 394, 401,
402, 405Kleidman, P. B. 172, 405Klemm, M. 70, 75, 230, 405Klingen, H. 261, 405Kneser, M. 304, 405Knill, E. 376, 381, 382, 392, 405Knus, M.-A. 12, 405Koch, H. 405Kogiso, T. 265, 405Kohnen, W. 251, 405Kondo, T. 405Kostrikin, A. I. vii, 405Koukouvinos, C. 345, 393, 399Krasikov, I. 326, 405Krawtchouk polynomial 42Kschischang, F. R. 337, 406Kunzer, M. xiiiKumar, P. V. viii, xii, 1, 51, 70, 73,
326, 395, 401, 402, 411
Laflamme, R. 376, 377, 381, 382, 405,411
Lam, C. W. H. 332, 335, 340, 342, 403,406
Lam, T. Y. 87–90, 97, 102, 117, 120,136, 406
λ, structure map 12Landjev, I. 151, 403Lang, S. 7, 406lattice 252
2-integral 260Barnes-Wall 171, 184
balanced 142, 185dual 252, 277E8 vii, 254even 252integral 252isotropic 277isotropic self-dual 277Leech vii, 254modular 2, 250Π-dual 255Type of 277unimodular 2, 278
Leeweight 30
weight enumerator 31Lee, Y. 345, 404Lee-extremal 317Leech, J. 406Lehner, J. 392length
of code 2of module 87
Leon, J. S. 70, 332, 338, 342, 345, 360,366, 367, 399, 406, 409
Leung, C.-M. xii, 415Leung, K. L. xiilevel 255
2-level 260even 253, 255
Levenshtein, V. I. 313, 328, 404, 406Li, S. 415Li, W.-C. W. 77Liebeck, M. W. 172, 405lifting to Z/4Z 74Ling, S. 244, 349, 393, 406van Lint, J. H. 1, 333, 339, 406Litsyn, S. 313, 326, 379, 381, 382, 392,
405, 406Loos, O. 8, 13, 406
M , twisted module 6mZ 53mZ
1 54mZ
II 54mZ
II,1 55mZ
S 55Ma, X. 319, 406MacLane, S. 105, 109, 112, 406MacWilliams
extension theorem 83, 89identity 35, 37–39, 253transform 139
MacWilliams, F. J. v, 1, 35, 89, 329,330, 333, 334, 339, 340, 349, 392,406
MAGMA xii, 35, 379, 385main theorems 150, 152, 164Mallows, C. L. v, x, 285, 294, 297, 318,
319, 324, 332, 337, 360, 367, 406,407
mapeven 3homogeneous 3
424 Index
pointed 3quadratic 3
Martinet, J. 252, 407Masley, J. M. 406mass formula 347Matn(R, M, ψ, Φ), matrix form ring
19Mat2(R, M, ψ, Φ), form ring 132Mat2(R, Φ), form ring 132Matm(R) 34Mathieu group 63, 67matrix
even 22generator 252Gram 252Hadamard 362
matrix form ring 19, 103Mattson, H. F., Jr. 80, 392McDonald, B. R. 51, 151, 407McEliece, R. J. 327, 382, 407McKean, H. P. 37, 398McLaughlin, S. W. xii, 398Mellinger, K. E. xii, 404Merkurjev, A. 405Mersereau, R. M. xii, 398Milnor, J. 252, 260, 407minimal distance 371
pure 371minimal injective cogenerator 88Miyake, T. 257, 258, 407Miyamoto, I. 405modular lattice 250modularity 258module
generator 116injective 87projective 116reflexive 87twisted 6
Molienseries vi, x, xii, 155
harmonic 162theorem 155
Molien, T. 155, 407monoid 113Moore, E. H. 334, 407Morita theory 116morphism 7, 103
of (R, S)-bimodules 113
of quadratic forms 114of quadratic groups 105weak 114
Mourrain, B. 40, 81, 234, 345, 394, 396Mumford, D. 263, 265, 266, 268, 407Munemasa, A. x, xiii, 326, 332, 339,
349, 351, 363, 393, 398, 402, 407
Nakayama, T. 407negative coefficients exist 319Nemenzo, F. R. 244, 349, 393Neumaier, A. 350, 395Nguyen, C. 334, 336, 393Nicholson, W. K. 89, 407Nielsen, M. A. 370, 372, 408Niemeier, H.-V. 283, 408Nilsson, J. E. M. xii, 397Nishimura, T. 332, 408Nobs, A. 302, 408Nochefranca, L. R. 394norm 252
Euclidean 30norm-extremal code 324notation xiv, 78
O’Meara, O. T. viii, 13, 131, 194, 401On(Fq), orthogonal group xvO-lattice 269octacode xii, 72, 73, 230–234, 237, 299oddity 256Odlyzko, A. M. 319, 324, 340, 406, 407optimal code xi, 313, 325order in form R-algebra 276orthogonal
geometry 350sum 15, 99
Ostergard, P. R. J. xiii, 339, 367, 394,401, 402, 408
Otmani, A. x, 331, 332, 338, 339, 395,399
Oura, M. x, 40, 70, 81, 172, 182, 184,324, 392, 398, 402, 408
Ozeki, M. x, 264, 332, 338, 391, 402,408
P (R, Φ), parabolic group 130p-Clifford
group 176p-excess 256
Index 425
p-signature 256(p)1, form ring 17(p)1, representation 17parabolic group 130Pareigis, B. 113, 408parity vector 25Parker, J. A. 332, 335, 403Parker, M. G. 339, 365, 397partial trace 370Pasquier, G. 332, 408Pasupathy, S. 337, 406Patterson, N. J. 212, 408Pauli operator 373Peres, A. 370, 408Perm(C) 24permutation group
of code 24permutation-equivalent
codes 23Pierce, J. 80Pierce, J. N. 330, 409Ping, L. xii, 408Piret, P. M. 1, 408Pittenger, A. O. 370, 408Playoust, C. 394Plesken, W. 272, 408Pless, V. S. x, xii, xiii, 1, 25, 30,
44, 64–67, 69, 70, 179, 280, 285,291, 294, 297, 313, 322, 326, 328,330–332, 334–342, 345, 349, 351,353, 354, 358, 360–363, 366, 367,392, 395, 396, 399, 404, 406–409
Poincare series 157, 305pointed
map 3representation 159
Poisson summation 37, 253, 266Poli, A. 334, 409polynomial invariant 131Poonen, B. 77positive definite 275
representation 272twisted algebra 272
positive semi-definite 274product
representation 100semidirect 130twisted algebra 100
progenerator 90, 116
faithfully balanced 90
projective
module 116
plane 362
representation 121, 140
Witt group 122
Witt ring 122
promotion xiv, 34
PSK 1, 186
pure code 371
qE 43, 46
qEII 44
qH 47
qH1 48
qH+ 49, 50
qH+1 49, 51
qH+II 50
qH+II,1 50
Qian, Z. 345, 409
Qk(k) 115
Qk, functor 115
qmodule 4
Quad, category 105
Quad-ring 112
Quad0(V, A), pointed maps 3
quadratic
form 114
group 104
k-algebra 116
map 3
pair 12
over Z 104
representation of 13
ring 112
quantum code 371
additive 374
binary 373
quasi-chain ring v, ix, 151
quasi-Frobenius ring 89
quasisimple
form ring 193
qubit 373, 376
Quebbemann, H.-G. xiii, 2, 44, 45, 224,250, 255, 276, 339, 407, 410
quotient 15
representation 99
426 Index
R(2I), form ring for Type I codes 16,41
R(2II), form ring for Type II codes 16,41
(R, M, ψ, Φ), form ring 13(R, Φ), form ring 13(R, M, ψ), twisted ring 6(R, S)-bimodule 113R − Mod − S category 113radical 15, 86Raev, R. V. 334, 413rate 30Ray-Chaudhuri, D. K. 349, 410Reed, I. S. xii, 410van Rees, G. H. J. xiii, 351, 393reflexive module 87Reiner, I. 137relative invariant 159representation
anisotropic 124conjugate 15, 100faithful 14finite 6, 13metabolic 121of form ring 13of quadratic pair 13of triangular form ring 18of triangular twisted ring 10of twisted module 6of twisted ring 7orthogonal sum of 15pointed 159positive definite 272product 100projective 121, 140quotient 99(T (V ), T (ρM ), β) 10(V, ρM ) 6(V, ρM , β) 7(V, ρM , ρΦ) 13(V, ρM , ρΦ, β) 13, 15
rescaled 9Reznick, B. 161, 162, 410ρ(2I) Type 16ρ(2II) Type 16Rifa, J. xii, 410Rigoni, C. 334, 409, 413ring xiv
chain v, 151
commutative 88form 13Frobenius 89Galois 51ground 2group 31invariant 155not Frobenius 89not quasi-Frobenius 89opposite 10quadratic 112quasi-chain v, ix, 151quasi-Frobenius 89self-injective 89semiperfect 136triangular twisted 9twisted 6
Rodemich, E. R., Jr. 327, 382, 407Room, T. G. vii, x, 171, 394Rost, M. 405Rowen, L. H. 89, 139, 410RS4 64–66, 222, 237, 338, 339, 363Rumsey, H. 327, 382, 407Runge, B. x, 172, 176, 184, 263–265,
410Ruseva, R. P. 332, 334, 410, 411, 415
Sφ shadow 24Sack, R. A. 318, 411Sah, C. H. 172, 411Salvati Manni, R. 251, 405Samorodnitsky, A. 325, 411scale 258Schafer, R. W. xii, 398Scharlau, R. 313, 337, 396, 411Scharlau, W. 13, 122, 127, 252, 283,
332, 336, 411Schmid, P. 172, 411Schmidt, S. E. 73, 366, 398, 400Schoeneberg, B. 279, 411Schomaker, D. 332, 336, 411Schulze-Pillot, R. 313, 411Segre, B. 349, 411Seidel, J. J. 161, 162, 172, 395, 397, 399Selberg trace formula viiself-complementary 65self-dual code 4, 6
over Fp 16quantum 371
Index 427
self-dual lattice 277self-glue 282self-injective 89self-orthogonal code 4, 6, 285, 367semidirect product 130semilinear 187semilinear similarity 115semisimple
form ring 14Senkevitch, N. I. 401sequence
A000027 210A000601 198, 218A001399 169, 205, 218, 224A001400 199, 222A001647 364A002623 217, 243A003178 352, 355A003179 352A004652 232A004657 232A005232 218A007979 234A007980 210, 233A008619 182, 217A008620 183, 206, 209, 223A008621 178, 271A008642 218A008647 221A008669 225A008670 204A008672 212, 270A008718 178, 180A008763 199A008769 221A014126 223A016729 341A020702 225A024186 180, 182A027633 184A027674 184A028249 203A028288 184A028309 222A028310 222A028344 212A028345 213A036410 205A039946 184
A051354 184A051462 233A052365 215A066016 342A066017 342A069247 226A090176 179A090899 365A092069 210A092070 211A092071 211A092072 211A092076 209A092091 215A092201 219A092203 219A092351 200A092352 203A092353 204A092354 206A092355 206A092496 220A092497 220A092498 223A092508 233A092531 231A092532 231A092533 231A092535 232A092544 235A092545 235A092546 235A092547 235A092548 246A092549 246A094927 365A097913 264A097950 213A097992 271A099595 241A099720 239A099748 240A099750 240A099752 240A099757 241A099770 242A100023 242A100024 243A100025 243
428 Index
A104993 200A105319 201A105510 363A105674 333A105675 333A105676 337A105677 338A105678 340A105681 342A105682 342A105685 333A105686 340A105687 341A105688 342A105689 342A106158 363A106159 363A106160 345A106161 345A106162 352A106163 352A106164 352A106165 352A106166 352A106167 352A106169 342A110160 180, 182A110193 352A110302 365A110306 365A110868 180A110869 180A110876 180A110880 180
seriesEisenstein 250Poincare 157
Seroussi, G. 338, 397Serre, J.-P. 36, 252, 253, 260, 411sextet group 354Seymour, P. D. 161, 411shadow 25, 260, 292, 317, 320
φ-shadow 24, 39extremal 326generalized 384pairs 26
Shen, A. H. 370, 404Shephard, G. C. 157, 411Shimura, G. 266, 411
Shin, D.-J. 326, 411Shiromoto, K. 89, 403, 411Shor, P. W. x, xi, 1, 26, 172, 173, 341,
365, 369, 372, 375–377, 379, 380,385–387, 395, 410, 411
shortening 387Siap, I. 411Sidelnikov, V. M. vi, x, 172, 412Siegel
half-plane 262theta series 262
Siegel C. L. 324Siegel, C. L. 261, 412signature 256similitude 84Simonis, J. 412simple
form ring 14, 193twisted ring 9
Singleton bound 381singly-even
code 41singly-even code 40Skoruppa, N. P. 304, 306, 398, 412Smith, L. 155, 157, 289, 412Smolin, J. A. 172, 392Snover, S. L. 64, 412Sobolev, S. L. 162, 412Sole, P. viii, x, xii, xiii, 1, 30, 40, 44, 66,
70, 73, 81, 234, 322, 326, 336, 339,342, 345, 349, 363, 391, 394–396,398, 399, 401, 402, 406, 409
Solomon, G. xii, 412Sp2n(Fq),symplectic group xvSpence, E. 412spherical design 161, 172, 181split type 195stabilizer code 375Stanley, R. P. 155, 157, 289, 413Steane, A. M. 372, 386, 413Stolze, J. 370, 413Storme, L. 89, 411strength 161strictly Type I 40structure map 12Sturmfels, B. 155, 413sub-Type 17, 372, 384subcode
maximal isotropic 24
Index 429
subgroupdual 35, 85
subquotient 98subtraction 289, 359Sun, F.-W. xii, 391Suter, D. 370, 413swe 31sweρ(C) 32sweρ
m 33Swiercz, S. 406symmetric
idempotent 101, 136symmetrized weight enumerator 31
ρ 32symplectic
geometry 350group 262
syzygy 156
T (M), triangular twisted ring 9t4 67, 209, 293, 337, 338, 361–363Tanabe, K. 326, 402, 413Tapia-Recillas, H. 73, 413τ , twist map 6, 104Taylor, M. J. 189, 399tensor product xiv, 34, 109tensor product of representations 101Terras, A. 37, 228, 413tetrad 366theorem
Assmus-Mattson 326Burmann-Lagrange 318Gleason vii, 178, 183, 206, 209, 291,
293Gleason-Pierce 1, 80, 326Hall 208Hecke viiHilbert 90 195Molien vii, 155Skolem-Noether 194
theta group 229, 249, 263, 279theta series 252, 278
average 251, 331Hilbert 270Jacobi-Siegel 265Riemann 265Siegel 262vector-valued 301
theta-group 279
Thiel, L. H. 332, 335, 403, 406Thompson, J. G. 270, 329, 330, 349,
406, 409, 413Tiep, P. H. vii, 191, 401, 405Tignol, J.-P. 405van Tilborg, H. C. A. 391Todd, J. A. 157, 411Tonchev, V. D. 326, 332, 334, 345, 395,
402–404, 409, 412, 413totally singular subspace 350triangular form ring 18triangular twisted ring 9Trott, M. D. 73, 399Truong, T. K. xii, 410Tsai, H.-P. 332, 413Tsushima, K. 265, 405Turyn, R. J. 80, 392twist map 6, 104twisted algebra 94, 272
positive definite 272product 100
twisted module 6representation of 6
twisted ring 6representation of 7rescaled 9simple 9
Typeof code 15of lattice 277sub- 17, 372, 384
Type Icode 16, 40, 41lattice 279
Type IIcode 5, 16, 40, 41lattice 279
Type IIIcode 67
Typesexamples of 60list of 40
U(f, I, Γ ), co-unitary group 131U(f,R), co-unitary group 131U(f, R, Φ), co-unitary group 131Un(Fq2), unitary group xvU(R, Φ), hyperbolic co-unitary group
133
430 Index
Um(R, Φ), hyperbolic co-unitary groupof degree m 136
unimodular 278unit
associated 6unitary geometry 350Uspensky, J. V. 74, 413
V , Hom(V, Q/Z) 10van Tilborg, H. C. A. xiiVarbanov, Z. 390Vardy, A. xii, 391, 413vector invariant 131Vega, G. 73, 413Vellbinger, U. xii, 400Venkov, B. B. 172, 268, 283, 317, 332,
402, 405, 407, 413Ventou, M. 334, 413Viterbo, E. xii, 400Vyalyi, M. N. 370, 404
Wall, G. E. vii, x, 171, 392, 394, 413Wan, Z.-X. 70, 349, 414Ward, H. N. 80, 89, 320, 331, 334, 338,
340, 360, 362, 406, 409, 414Watson, G. N. 318, 414WAut(C) 24weak equivalence 23, 84weight
divisibility of 80Hamming 29Lee 30
weight enumerator 30, 377average 329biweight 33complete 30dual 377full 31genus-m ρ-symmetrized 33genus-m complete 33genus-m full 34Hamming 30higher genus 33Lee 31
multiple 33shadow 377symmetrized 31
Weight Enumerator Conjecture x,150, 163
Weil representation 302Weil, A. vii, 142, 145, 301, 414Welch, L. R. 327, 382, 407Whittaker, E. T. 318, 414Winter, D. L. 172, 177, 208, 414Witt
-equivalent 123group 103, 123, 287group, projective 122-null 123ring, projective 122vector 216
Wolfmann, J. xii, 414Wood, J. A. ix, 83, 89, 131, 414Wootters, W. K. 172, 392wreath product 23
Yankov, N. 332, 415Yeung, K. L. 408Yin, X. xii, 410Yorgov, V. Y. 332, 334, 336, 392, 394,
403, 413–415Yousif, M. F. 89, 407Yuan, J. xii, 415
Z/mZ, integers mod m xvZ/mZ-linear code 53z12 69, 341, 386Zn, cyclic group xvZn, n-adic integers xv, 60Zagier, D. 261, 275, 398van Zanter, A. 397Zaslavsky, T. 161, 411Zassenhaus, H. 121, 415Zhang, S. 319, 415Zhu, L. 319, 406Ziapkov, N. P. 332, 415Zinoviev, V. A. xii, 397Zurek, W. 376, 405